\r\n\t \r\n\tIn this book Advanced application of radionuclides are introduced. New global trends on safe application of radionuclides in human life is elucidated.
",isbn:"978-1-78985-984-3",printIsbn:"978-1-78985-356-8",pdfIsbn:"978-1-83880-081-9",doi:null,price:0,priceEur:0,priceUsd:0,slug:null,numberOfPages:0,isOpenForSubmission:!0,hash:"3ec120ceb22cf08366724e87f8eaa649",bookSignature:"Dr. Ali Nabipour Chakoli",publishedDate:null,coverURL:"https://cdn.intechopen.com/books/images_new/8780.jpg",keywords:"Radiation Measurement, Radiation Effects on Cells, Radon and its Douthres, Uranium and its Douthres, Isotope Hydrology, Water Health, Nuclear Medicine, Agricultural Products Improvement, Trace elements, Nutrition Enhancement, Radiographic test, Leakage tests",numberOfDownloads:null,numberOfWosCitations:0,numberOfCrossrefCitations:0,numberOfDimensionsCitations:null,numberOfTotalCitations:null,isAvailableForWebshopOrdering:!0,dateEndFirstStepPublish:"November 20th 2019",dateEndSecondStepPublish:"December 11th 2019",dateEndThirdStepPublish:"February 9th 2020",dateEndFourthStepPublish:"April 29th 2020",dateEndFifthStepPublish:"June 28th 2020",remainingDaysToSecondStep:"3 days",secondStepPassed:!1,currentStepOfPublishingProcess:2,editedByType:null,kuFlag:!1,editors:[{id:"250668",title:"Dr.",name:"Ali",middleName:null,surname:"Nabipour Chakoli",slug:"ali-nabipour-chakoli",fullName:"Ali Nabipour Chakoli",profilePictureURL:"https://mts.intechopen.com/storage/users/250668/images/system/250668.jpg",biography:"Academic Qualification:\r\n•\tPhD in Materials Physics and Chemistry, From: Sep. 2006, to: Sep. 2010, School of Materials Science and Engineering, Harbin Institute of Technology, Thesis: Structure and Shape Memory Effect of Functionalized MWCNTs/poly (L-lactide-co-ε-caprolactone) Nanocomposites. Supervisor: Prof. Wei Cai,\r\n•\tM.Sc in Applied Physics, From: 1996, to: 1998, Faculty of Physics & Nuclear Science, Amirkabir Uni. of Technology, Tehran, Iran, Thesis: Determination of Boron in Micro alloy Steels with solid state nuclear track detectors by neutron induced auto radiography, Supervisors: Dr. M. Hosseini Ashrafi and Dr. A. Hosseini.\r\n•\tB.Sc. in Applied Physics, From: 1991, to: 1996, Faculty of Physics & Nuclear Science, Amirkabir Uni. of Technology, Tehran, Iran, Thesis: Design of shielding for Am-Be neutron sources for In Vivo neutron activation analysis, Supervisor: Dr. M. Hosseini Ashrafi.\r\n\r\nResearch Experiences:\r\n1.\tNanomaterials, Carbon Nanotubes, Graphene: Synthesis, Functionalization and Characterization,\r\n2.\tMWCNTs/Polymer Composites: Fabrication and Characterization, \r\n3.\tShape Memory Polymers, Biodegradable Polymers, ORC, Collagen,\r\n4.\tMaterials Analysis and Characterizations: TEM, SEM, XPS, FT-IR, Raman, DSC, DMA, TGA, XRD, GPC, Fluoroscopy, \r\n5.\tInteraction of Radiation with Mater, Nuclear Safety and Security, NDT(RT),\r\n6.\tRadiation Detectors, Calibration (SSDL),\r\n7.\tCompleted IAEA e-learning Courses:\r\nNuclear Security (15 Modules),\r\nNuclear Safety:\r\nTSA 2: Regulatory Protection in Occupational Exposure,\r\nTips & Tricks: Radiation Protection in Radiography,\r\nSafety and Quality in Radiotherapy,\r\nCourse on Sealed Radioactive Sources,\r\nCourse on Fundamentals of Environmental Remediation,\r\nCourse on Planning for Environmental Remediation,\r\nKnowledge Management Orientation Course,\r\nFood Irradiation - Technology, Applications and Good Practices,\r\nEmployment:\r\nFrom 2010 to now: Academic staff, Nuclear Science and Technology Research Institute, Kargar Shomali, Tehran, Iran, P.O. Box: 14395-836.\r\nFrom 1997 to 2006: Expert of Materials Analysis and Characterization. Research Center of Agriculture and Medicine. Rajaeeshahr, Karaj, Iran, P. O. Box: 31585-498.",institutionString:"Atomic Energy Organization of Iran",position:null,outsideEditionCount:0,totalCites:0,totalAuthoredChapters:"1",totalChapterViews:"0",totalEditedBooks:"0",institution:{name:"Atomic Energy Organization of Iran",institutionURL:null,country:{name:"Iran"}}}],coeditorOne:null,coeditorTwo:null,coeditorThree:null,coeditorFour:null,coeditorFive:null,topics:[{id:"20",title:"Physics",slug:"physics"}],chapters:null,productType:{id:"1",title:"Edited Volume",chapterContentType:"chapter",authoredCaption:"Edited by"},personalPublishingAssistant:{id:"225753",firstName:"Marina",lastName:"Dusevic",middleName:null,title:"Ms.",imageUrl:"https://mts.intechopen.com/storage/users/225753/images/7224_n.png",email:"marina.d@intechopen.com",biography:"As an Author Service Manager my responsibilities include monitoring and facilitating all publishing activities for authors and editors. From chapter submission and review, to approval and revision, copyediting and design, until final publication, I work closely with authors and editors to ensure a simple and easy publishing process. I maintain constant and effective communication with authors, editors and reviewers, which allows for a level of personal support that enables contributors to fully commit and concentrate on the chapters they are writing, editing, or reviewing. I assist authors in the preparation of their full chapter submissions and track important deadlines and ensure they are met. I help to coordinate internal processes such as linguistic review, and monitor the technical aspects of the process. As an ASM I am also involved in the acquisition of editors. Whether that be identifying an exceptional author and proposing an editorship collaboration, or contacting researchers who would like the opportunity to work with IntechOpen, I establish and help manage author and editor acquisition and contact."}},relatedBooks:[{type:"book",id:"3161",title:"Frontiers in Guided Wave Optics and Optoelectronics",subtitle:null,isOpenForSubmission:!1,hash:"deb44e9c99f82bbce1083abea743146c",slug:"frontiers-in-guided-wave-optics-and-optoelectronics",bookSignature:"Bishnu Pal",coverURL:"https://cdn.intechopen.com/books/images_new/3161.jpg",editedByType:"Edited by",editors:[{id:"4782",title:"Prof.",name:"Bishnu",surname:"Pal",slug:"bishnu-pal",fullName:"Bishnu Pal"}],productType:{id:"1",chapterContentType:"chapter",authoredCaption:"Edited by"}},{type:"book",id:"72",title:"Ionic Liquids",subtitle:"Theory, Properties, New Approaches",isOpenForSubmission:!1,hash:"d94ffa3cfa10505e3b1d676d46fcd3f5",slug:"ionic-liquids-theory-properties-new-approaches",bookSignature:"Alexander Kokorin",coverURL:"https://cdn.intechopen.com/books/images_new/72.jpg",editedByType:"Edited by",editors:[{id:"19816",title:"Prof.",name:"Alexander",surname:"Kokorin",slug:"alexander-kokorin",fullName:"Alexander Kokorin"}],productType:{id:"1",chapterContentType:"chapter",authoredCaption:"Edited by"}},{type:"book",id:"1591",title:"Infrared Spectroscopy",subtitle:"Materials Science, Engineering and Technology",isOpenForSubmission:!1,hash:"99b4b7b71a8caeb693ed762b40b017f4",slug:"infrared-spectroscopy-materials-science-engineering-and-technology",bookSignature:"Theophile Theophanides",coverURL:"https://cdn.intechopen.com/books/images_new/1591.jpg",editedByType:"Edited by",editors:[{id:"37194",title:"Dr.",name:"Theophanides",surname:"Theophile",slug:"theophanides-theophile",fullName:"Theophanides Theophile"}],productType:{id:"1",chapterContentType:"chapter",authoredCaption:"Edited by"}},{type:"book",id:"1373",title:"Ionic Liquids",subtitle:"Applications and Perspectives",isOpenForSubmission:!1,hash:"5e9ae5ae9167cde4b344e499a792c41c",slug:"ionic-liquids-applications-and-perspectives",bookSignature:"Alexander Kokorin",coverURL:"https://cdn.intechopen.com/books/images_new/1373.jpg",editedByType:"Edited by",editors:[{id:"19816",title:"Prof.",name:"Alexander",surname:"Kokorin",slug:"alexander-kokorin",fullName:"Alexander Kokorin"}],productType:{id:"1",chapterContentType:"chapter",authoredCaption:"Edited by"}},{type:"book",id:"4816",title:"Face Recognition",subtitle:null,isOpenForSubmission:!1,hash:"146063b5359146b7718ea86bad47c8eb",slug:"face_recognition",bookSignature:"Kresimir Delac and Mislav Grgic",coverURL:"https://cdn.intechopen.com/books/images_new/4816.jpg",editedByType:"Edited by",editors:[{id:"528",title:"Dr.",name:"Kresimir",surname:"Delac",slug:"kresimir-delac",fullName:"Kresimir Delac"}],productType:{id:"1",chapterContentType:"chapter",authoredCaption:"Edited by"}},{type:"book",id:"57",title:"Physics and Applications of Graphene",subtitle:"Experiments",isOpenForSubmission:!1,hash:"0e6622a71cf4f02f45bfdd5691e1189a",slug:"physics-and-applications-of-graphene-experiments",bookSignature:"Sergey Mikhailov",coverURL:"https://cdn.intechopen.com/books/images_new/57.jpg",editedByType:"Edited by",editors:[{id:"16042",title:"Dr.",name:"Sergey",surname:"Mikhailov",slug:"sergey-mikhailov",fullName:"Sergey Mikhailov"}],productType:{id:"1",chapterContentType:"chapter",authoredCaption:"Edited by"}},{type:"book",id:"3092",title:"Anopheles mosquitoes",subtitle:"New insights into malaria vectors",isOpenForSubmission:!1,hash:"c9e622485316d5e296288bf24d2b0d64",slug:"anopheles-mosquitoes-new-insights-into-malaria-vectors",bookSignature:"Sylvie Manguin",coverURL:"https://cdn.intechopen.com/books/images_new/3092.jpg",editedByType:"Edited by",editors:[{id:"50017",title:"Prof.",name:"Sylvie",surname:"Manguin",slug:"sylvie-manguin",fullName:"Sylvie Manguin"}],productType:{id:"1",chapterContentType:"chapter",authoredCaption:"Edited by"}},{type:"book",id:"3794",title:"Swarm Intelligence",subtitle:"Focus on Ant and Particle Swarm Optimization",isOpenForSubmission:!1,hash:"5332a71035a274ecbf1c308df633a8ed",slug:"swarm_intelligence_focus_on_ant_and_particle_swarm_optimization",bookSignature:"Felix T.S. Chan and Manoj Kumar Tiwari",coverURL:"https://cdn.intechopen.com/books/images_new/3794.jpg",editedByType:"Edited by",editors:[{id:"252210",title:"Dr.",name:"Felix",surname:"Chan",slug:"felix-chan",fullName:"Felix Chan"}],productType:{id:"1",chapterContentType:"chapter",authoredCaption:"Edited by"}},{type:"book",id:"3621",title:"Silver Nanoparticles",subtitle:null,isOpenForSubmission:!1,hash:null,slug:"silver-nanoparticles",bookSignature:"David Pozo Perez",coverURL:"https://cdn.intechopen.com/books/images_new/3621.jpg",editedByType:"Edited by",editors:[{id:"6667",title:"Dr.",name:"David",surname:"Pozo",slug:"david-pozo",fullName:"David Pozo"}],productType:{id:"1",chapterContentType:"chapter",authoredCaption:"Edited by"}},{type:"book",id:"371",title:"Abiotic Stress in Plants",subtitle:"Mechanisms and Adaptations",isOpenForSubmission:!1,hash:"588466f487e307619849d72389178a74",slug:"abiotic-stress-in-plants-mechanisms-and-adaptations",bookSignature:"Arun Shanker and B. Venkateswarlu",coverURL:"https://cdn.intechopen.com/books/images_new/371.jpg",editedByType:"Edited by",editors:[{id:"58592",title:"Dr.",name:"Arun",surname:"Shanker",slug:"arun-shanker",fullName:"Arun Shanker"}],productType:{id:"1",chapterContentType:"chapter",authoredCaption:"Edited by"}}]},chapter:{item:{type:"chapter",id:"66332",title:"Polynomials with Symmetric Zeros",doi:"10.5772/intechopen.82728",slug:"polynomials-with-symmetric-zeros",body:'\n
\n
1. Introduction
\n
In this work, we consider the theory of self-conjugate (SC), self-reciprocal (SR), and self-inversive (SI) polynomials. These are polynomials whose zeros are symmetric either to the real line \n\nR\n\n or to the unit circle \n\nS\n=\n\n\nz\n∈\nC\n:\n\nz\n\n=\n1\n\n\n\n. The basic properties of these polynomials can be found in the books of Marden [1], Milovanović et al. [2], and Sheil-Small [3]. Although these polynomials are very important in both mathematics and physics, it seems that there is no specific review about them; in this work, we present a bird’s eye view to this theory, focusing on the zeros of such polynomials. Other properties of these polynomials (e.g., irreducibility, norms, analytical properties, etc.) are not covered here due to short space, nonetheless, the interested reader can check many of the references presented in the bibliography to this end.
\n
\n
\n
2. Self-conjugate, self-reciprocal, and self-inversive polynomials
\n
We begin with some definitions:
\n
Definition 1. Let \n\np\n\nz\n\n=\n\np\n0\n\n+\n\np\n1\n\nz\n+\n⋯\n+\n\np\n\nn\n−\n1\n\n\n\nz\n\nn\n−\n1\n\n\n+\n\np\nn\n\n\nz\nn\n\n\n be a polynomial of degree \n\nn\n\n with complex coefficients. We shall introduce three polynomials, namely the conjugate polynomial\n\n\np\n¯\n\n\nz\n\n\n, the reciprocal polynomial\n\n\np\n∗\n\n\nz\n\n\n, and the inversive polynomial\n\n\np\n†\n\n\nz\n\n\n, which are, respectively, defined in terms of \n\np\n\nz\n\n\n as follows:
where the bar means complex conjugation. Notice that the conjugate, reciprocal, and inversive polynomials can also be defined without making reference to the coefficients of \n\np\n\nz\n\n\n:
From these relations, we plainly see that if \n\n\nζ\n1\n\n,\n…\n,\n\nζ\nn\n\n\n are the zeros of a complex polynomial \n\np\n\nz\n\n\n of degree \n\nn\n\n, then, the zeros of \n\n\np\n¯\n\n\nz\n\n\n are \n\n\n\nζ\n1\n\n¯\n\n,\n…\n,\n\n\nζ\nn\n\n¯\n\n\n, the zeros of \n\n\np\n∗\n\n\nz\n\n\n are \n\n1\n/\n\nζ\n1\n\n,\n…\n,\n1\n/\n\nζ\nn\n\n\n, and finally, the zeros of \n\n\np\n†\n\n\nz\n\n\n are \n\n1\n/\n\n\nζ\n1\n\n¯\n\n,\n…\n,\n1\n/\n\n\nζ\nn\n\n¯\n\n\n. Thus, if \n\np\n\nz\n\n\n has \n\nk\n\n zeros on \n\nR\n\n, \n\nl\n\n zeros on the upper half-plane \n\n\nC\n+\n\n=\n\n\nz\n∈\nC\n:\nIm\n\nz\n\n>\n0\n\n\n\n, and \n\nm\n\n zeros in the lower half-plane \n\n\nC\n−\n\n=\n\n\nz\n∈\nC\n:\nIm\n\nz\n\n<\n0\n\n\n\n so that \n\nk\n+\nl\n+\nm\n=\nn\n\n, then \n\n\np\n¯\n\n\nz\n\n\n will have the same number \n\nk\n\n of zeros on \n\nR\n\n, \n\nl\n\n zeros in \n\n\nC\n−\n\n\n and \n\nm\n\n zeros in \n\n\nC\n+\n\n\n. Similarly, if \n\np\n\nz\n\n\n has \n\nk\n\n zeros on \n\nS\n\n, \n\nl\n\n zeros inside \n\nS\n\n and \n\nm\n\n zeros outside \n\nS\n\n, so that \n\nk\n+\nl\n+\nm\n=\nn\n\n, then both \n\n\np\n∗\n\n\nz\n\n\n as \n\n\np\n†\n\n\nz\n\n\n will have the same number \n\nk\n\n of zeros on \n\nS\n\n, \n\nl\n\n zeros outside \n\nS\n\n and \n\nm\n\n zeros inside \n\nS\n\n.
\n
These properties encourage us to introduce the following classes of polynomials:
\n
Definition 2. A complex polynomial \n\np\n\nz\n\n\n is called1self-conjugate (SC), self-reciprocal (SR), or self-inversive (SI) if, for any zero \n\nζ\n\n of \n\np\n\nz\n\n\n, the complex-conjugate \n\n\nζ\n¯\n\n\n, the reciprocal \n\n1\n/\nζ\n\n, or the reciprocal of the complex-conjugate \n\n1\n/\n\nζ\n¯\n\n\n is also a zero of \n\np\n\nz\n\n\n, respectively.
\n
Thus, the zeros of any SC polynomial are all symmetric to the real line \n\nR\n\n, while the zeros of the any SI polynomial are symmetric to the unit circle \n\nS\n\n. The zeros of any SR polynomial are obtained by an inversion with respect to the unit circle followed by a reflection in the real line. From this, we can establish the following:
\n
Theorem 1.If\n\np\n\nz\n\n\nis an SC polynomial of odd degree, then it necessarily has at least one zero on\n\nR\n\n. Similarly, if\n\np\n\nz\n\n\nis an SR or SI polynomial of odd degree, then it necessarily has at least one zero on\n\nS\n\n.
\n
Proof. From Definition 2 it follows that the number of non-real zeros of an SC polynomial \n\np\n\nz\n\n\n can only occur in (conjugate) pairs; thus, if \n\np\n\nz\n\n\n has odd degree, then at least one zero of it must be real. Similarly, the zeros of \n\n\np\n†\n\n\nz\n\n\n or \n\n\np\n∗\n\n\nz\n\n\n that have modulus different from \n\n1\n\n can only occur in (inversive or reciprocal) pairs as well; thus, if \n\np\n\nz\n\n\n has odd degree then at least one zero of it must lie on \n\nS\n\n.□
\n
Theorem 2.The necessary and sufficient condition for a complex polynomial\n\np\n\nz\n\n\nto be SC, SR, or SI is that there exists a complex number\n\nω\n\nof modulus\n\n1\n\nso that one of the following relations, respectively, holds:
Proof. It is clear in view of (1) and (2) that these conditions are sufficient. We need to show, therefore, that these conditions are also necessary. Let us suppose first that \n\np\n\nz\n\n\n is SC. Then, for any zero \n\nζ\n\n of \n\np\n\nz\n\n\n the complex-conjugate number \n\n\nζ\n¯\n\n\n is also a zero of it. Thus, we can write
with \n\nω\n=\n\np\nn\n\n/\n\n\np\nn\n\n¯\n\n\n so that \n\n∣\nω\n∣\n=\n\n\n\np\nn\n\n/\n\n\np\nn\n\n¯\n\n\n\n=\n1\n\n. Now, let us suppose that \n\np\n\nz\n\n\n is SR. Then, for any zero \n\nζ\n\n of \n\np\n\nz\n\n\n, the reciprocal number \n\n1\n/\nζ\n\n is also a zero of it; thus,
with \n\nω\n=\n\n\n\n−\n1\n\n\nn\n\n/\n\n\n\nζ\n1\n\n…\n\nζ\nn\n\n\n\n=\n\np\nn\n\n/\n\np\n0\n\n\n; now, for any zero \n\nζ\n\n of \n\np\n\nz\n\n\n (which is necessarily different from zero if \n\np\n\nz\n\n\n is SR), there will be another zero whose value is \n\n1\n/\nζ\n\n so that \n\n\n\n\nζ\n1\n\n…\n\nζ\nn\n\n\n\n=\n1\n\n, which implies \n\n∣\nω\n∣\n=\n1\n\n. The proof for SI polynomials is analogous and will be concealed; it follows that \n\nω\n=\n\np\nn\n\n/\n\n\np\n0\n\n¯\n\n\n in this case.□
\n
Now from (1), (2) and (3), we can conclude that the coefficients of an SC, an SR, and an SI polynomial of degree \n\nn\n\n satisfy, respectively, the following relations:
We highlight that any real polynomial is SC—in fact, many theorems which are valid for real polynomials are also valid for, or can be easily extended to, SC polynomials.
\n
There also exist polynomials whose zeros are symmetric with respect to both the real line \n\nR\n\n and the unit circle \n\nS\n\n. A polynomial \n\np\n\nz\n\n\n with this double symmetry is, at the same time, SC and SI (and, hence, SR as well). This is only possible if all the coefficients of \n\np\n\nz\n\n\n are real, which implies that \n\nω\n=\n±\n1\n\n. This suggests the following additional definitions:
\n
Definition 3. A real self-reciprocal polynomial \n\np\n\nz\n\n\n that satisfies the relation \n\np\n\nz\n\n=\nω\n\nz\nn\n\np\n\n\n1\n/\nz\n\n\n\n will be called a positive self-reciprocal (PSR) polynomial if \n\nω\n=\n1\n\n and a negative self-reciprocal (NSR) polynomial if \n\nω\n=\n−\n1\n\n.
\n
Thus, the coefficients of any PSR polynomial \n\np\n\nz\n\n=\n\np\n0\n\n+\n⋯\n+\n\np\nn\n\n\nz\nn\n\n\n of degree \n\nn\n\n satisfy the relations \n\n\np\nk\n\n=\n\np\n\nn\n−\nk\n\n\n\n for \n\n0\n⩽\nk\n⩽\nn\n\n, while the coefficients of any NSR polynomial \n\np\n\nz\n\n\n of degree \n\nn\n\n satisfy the relations \n\n\np\nk\n\n=\n−\n\np\n\nn\n−\nk\n\n\n\n for \n\n0\n⩽\nk\n⩽\nn\n\n; this last condition implies that the middle coefficient of an NSR polynomial of even degree is always zero.
\n
Some elementary properties of PSR and NSR polynomials are the following: first, notice that, if \n\nζ\n\n is a zero of any PSR or NSR polynomial \n\np\n\nz\n\n\n of degree \n\nn\n⩾\n4\n\n, then the three complex numbers \n\n1\n/\nζ\n\n, \n\n\nζ\n¯\n\n\n and \n\n1\n/\n\nζ\n¯\n\n\n are also zeros of \n\np\n\nz\n\n\n. In particular, the number of zeros of such polynomials which are neither in \n\nS\n\n or in \n\nR\n\n is always a multiple of \n\n4\n\n. Besides, any NSR polynomial has \n\nz\n=\n1\n\n as a zero and \n\np\n\nz\n\n/\n\n\nz\n−\n1\n\n\n\n is PSR; further, if \n\np\n\nz\n\n\n has even degree then \n\nz\n=\n−\n1\n\n is also a zero of it and \n\np\n\nz\n\n/\n\n\n\nz\n2\n\n−\n1\n\n\n\n is a PSR polynomial of even degree. In a similar way, any PSR polynomial \n\np\n\nz\n\n\n of odd degree has \n\nz\n=\n−\n1\n\n as a zero and \n\np\n\nz\n\n/\n\n\nz\n+\n1\n\n\n\n is also PSR. The product of two PSR, or two NSR, polynomials is PSR, while the product of a PSR polynomial with an NSR polynomial is NSR. These statements follow directly from the definitions of such polynomials.
\n
We also mention that any PSR polynomial of even degree (say, \n\nn\n=\n2\nm\n\n) can be written in the following form:
an expression that is obtained by using the relations \n\n\np\nk\n\n=\n\np\n\n2\nm\n−\nk\n\n\n\n, \n\n0\n⩽\nk\n⩽\n2\nm\n\n, and gathering the terms of \n\np\n\nz\n\n\n with the same coefficients. Furthermore, the expression \n\n\nZ\ns\n\n\nz\n\n=\n\n\n\nz\ns\n\n+\n\nz\n\n−\ns\n\n\n\n\n\n for any integer \n\ns\n\n can be written as a polynomial of degree \n\ns\n\n in the new variable \n\nx\n=\nz\n+\n1\n/\nz\n\n (the proof follows easily by induction over \n\ns\n\n); thus, we can write \n\np\n\nz\n\n=\n\nz\nm\n\nq\n\nx\n\n\n, where \n\nq\n\nx\n\n=\n\nq\n0\n\n+\n⋯\n+\n\nq\nm\n\n\nx\nm\n\n\n is such that the coefficients \n\n\nq\n0\n\n,\n…\n,\n\nq\nm\n\n\n are certain functions of \n\n\np\n0\n\n,\n…\n,\n\np\nm\n\n\n. From this we can state the following:
\n
Theorem 3.Let\n\np\n\nz\n\n\nbe a PSR polynomial of even degree\n\nn\n=\n2\nm\n\n. For each pair\n\nζ\n\nand\n\n1\n/\nζ\n\nof self-reciprocal zeros of\n\np\n\nz\n\n\nthat lie on\n\nS\n\n, there is a corresponding zero\n\nξ\n\nof the polynomial\n\nq\n\nx\n\n\n, as defined above, in the interval\n\n\n\n−\n2\n\n2\n\n\nof the real line.
\n
Proof. For each zero \n\nζ\n\n of \n\np\n\nx\n\n\n that lie on \n\nS\n\n, write \n\nζ\n=\n\ne\niθ\n\n\n for some \n\nθ\n∈\nR\n\n. Thereby, as \n\nq\n\nx\n\n=\nq\n\n\nz\n+\n1\n/\nz\n\n\n=\np\n\nz\n\n/\n\nz\nm\n\n\n, it follows that \n\nξ\n=\nζ\n+\n1\n/\nζ\n=\n2\ncos\nθ\n\n will be a zero of \n\nq\n\nx\n\n\n. This shows us that \n\nξ\n\n is limited to the interval \n\n\n\n−\n2\n\n2\n\n\n of the real line. Finally, notice that the reciprocal zero \n\n1\n/\nζ\n\n of \n\np\n\nz\n\n\n is mapped to the same zero \n\nξ\n\n of \n\nq\n\nx\n\n\n.□
\n
Finally, remembering that the Chebyshev polynomials of first kind, \n\n\nT\nn\n\n\nz\n\n\n, are defined by the formula \n\n\nT\nn\n\n\n\n\n1\n2\n\n\n\nz\n+\n\nz\n\n−\n1\n\n\n\n\n\n\n=\n\n1\n2\n\n\n\n\nz\nn\n\n+\n\nz\n\n−\nn\n\n\n\n\n\n for \n\nz\n∈\nC\n\n, it follows as well that \n\nq\n\nx\n\n\n, and hence any PSR polynomial, can be written as a linear combination of Chebyshev polynomials:
3. How these polynomials are related to each other?
\n
In this section, we shall analyze how SC, SR, and SI polynomials are related to each other. Let us begin with the relationship between the SR and SI polynomials, which is actually very simple: indeed, from (1), (2), and (3) we can see that each one is nothing but the conjugate polynomial of the other, that is
Thus, if \n\np\n\nz\n\n\n is an SR (SI) polynomial, then \n\n\np\n¯\n\n\nz\n\n\n will be SI (SR) polynomial. Because of this simple relationship, several theorems which are valid for SI polynomials are also valid for SR polynomials and vice versa.
\n
The relationship between SC and SI polynomials is not so easy to perceive. A way of revealing their connection is to make use of a suitable pair of Möbius transformations, that maps the unit circle onto the real line and vice versa, which is often called Cayley transformations, defined through the formulas:
This approach was developed in [4], where some algorithms for counting the number of zeros that a complex polynomial has on the unit circle were also formulated.
\n
It is an easy matter to verify that \n\nM\n\nz\n\n\n maps \n\nR\n\n onto \n\nS\n\n while \n\nW\n\nz\n\n\n maps \n\nS\n\n onto \n\nR\n\n. Besides, \n\nM\n\nz\n\n\n maps the upper (lower) half-plane to the interior (exterior) of \n\nS\n\n, while \n\nW\n\nz\n\n\n maps the interior (exterior) of \n\nS\n\n onto the upper (lower) half-plane. Notice that \n\nW\n\nz\n\n\n can be thought as the inverse of \n\nM\n\nz\n\n\n in the Riemann sphere \n\n\nC\n∞\n\n=\nC\n∪\n\n∞\n\n\n, if we further assume that \n\nM\n\n\n−\ni\n\n\n=\n∞\n\n, \n\nM\n\n∞\n\n=\n1\n\n, \n\nW\n\n1\n\n=\n∞\n\n, and \n\nW\n\n∞\n\n=\n−\ni\n\n.
\n
Given a polynomial \n\np\n\nz\n\n\n of degree \n\nn\n\n, we define two Möbius-transformed polynomials, namely
The following theorem shows us how the zeros of \n\nQ\n\nz\n\n\n and \n\nT\n\nz\n\n\n are related with the zeros of \n\np\n\nz\n\n\n:
\n
Theorem 4.Let\n\n\nζ\n1\n\n,\n…\n,\n\nζ\nn\n\n\ndenote the zeros of\n\np\n\nz\n\n\nand\n\n\nη\n1\n\n,\n…\n,\n\nη\nn\n\n\nthe respective zeros of\n\nQ\n\nz\n\n\n. Provided\n\np\n\n1\n\n≠\n0\n\n, we have that\n\n\nη\n1\n\n=\nW\n\n\nζ\n1\n\n\n,\n…\n,\n\nη\nn\n\n=\nW\n\n\nζ\nn\n\n\n\n. Similarly, if\n\n\nτ\n1\n\n,\n…\n\nτ\nn\n\n\nare the zeros of\n\nT\n\nz\n\n\n, then we have\n\n\nτ\n1\n\n=\nM\n\n\nζ\n1\n\n\n,\n…\n,\n\nτ\nn\n\n=\nM\n\n\nζ\nn\n\n\n\n, provided that\n\np\n\n\n−\ni\n\n\n≠\n0\n\n.
\n
Proof. In fact, inverting the expression for \n\nQ\n\nz\n\n\n and evaluating it in any zero \n\n\nζ\nk\n\n\n of \n\np\n\nz\n\n\n we get that \n\np\n\n\nζ\nk\n\n\n=\n\n\n\n−\ni\n/\n2\n\n\nn\n\n\n\n\n\nζ\nk\n\n−\n1\n\n\nn\n\nQ\n\n\nW\n\n\nζ\nk\n\n\n\n\n=\n0\n\n for \n\n0\n⩽\nk\n\n⩽\nn\n\n. Provided that \n\nz\n=\n1\n\n is not a zero of \n\np\n\nz\n\n\n we get that \n\n\nη\nk\n\n=\nW\n\n\nζ\nk\n\n\n\n is a zero of \n\nQ\n\nz\n\n\n. The proof for the zeros of \n\nT\n\nz\n\n\n is analogous.□
\n
This result also shows that \n\nQ\n\nz\n\n\n and \n\nT\n\nz\n\n\n have the same degree as \n\np\n\nz\n\n\n whenever \n\np\n\n1\n\n≠\n0\n\n or \n\np\n\n\n−\ni\n\n\n≠\n0\n\n, respectively. In fact, if \n\np\n\nz\n\n\n has a zero at \n\nz\n=\n1\n\n of multiplicity \n\nm\n\n then \n\nQ\n\nz\n\n\n will be a polynomial of degree \n\nn\n−\nm\n\n, the same being true for \n\nT\n\nz\n\n\n if \n\np\n\nz\n\n\n has a zero of multiplicity \n\nm\n\n at \n\nz\n=\n−\ni\n\n. This can be explained by the fact that the points \n\nz\n=\n1\n\n and \n\nz\n=\n−\ni\n\n are mapped to infinity by \n\nW\n\nz\n\n\n and \n\nM\n\nz\n\n\n, respectively.
\n
The following theorem shows that the set of SI polynomials are isomorphic to the set of SC polynomials:
\n
Theorem 5.Let\n\np\n\nz\n\n\nbe an SI polynomial. Then, the transformed polynomial\n\nQ\n\nz\n\n=\n\n\n\nz\n+\ni\n\n\nn\n\np\n\n\nM\n\nz\n\n\n\n\nis an SC polynomial. Similarly, if\n\np\n\nz\n\n\nis an SC polynomial, then\n\nT\n\nz\n\n=\n\n\n\nz\n−\n1\n\n\nn\n\np\n\n\nW\n\nz\n\n\n\n\nwill be an SI polynomial.
\n
Proof. Let \n\nζ\n\n and \n\n1\n/\n\nζ\n¯\n\n\n be two inversive zeros an SI polynomial \n\np\n\nz\n\n\n. Then, according to Theorem 4, the corresponding zeros of \n\nQ\n\nz\n\n\n will be:
Thus, any pair of zeros of \n\np\n\nz\n\n\n that are symmetric to the unit circle are mapped in zeros of \n\nQ\n\nz\n\n\n that are symmetric to the real line; because \n\np\n\nz\n\n\n is SI, it follows that \n\nQ\n\nz\n\n\n is SC. Conversely, let \n\nζ\n\n and \n\n\nζ\n¯\n\n\n be two zeros of an SC polynomial \n\np\n\nz\n\n\n; then the corresponding zeros of \n\nT\n\nz\n\n\n will be:
Thus, any pair of zeros of \n\np\n\nz\n\n\n that are symmetric to the real line are mapped in zeros of \n\nT\n\nz\n\n\n that are symmetric to the unit circle. Because \n\np\n\nz\n\n\n is SC, it follows that \n\nT\n\nz\n\n\n is SI.□
\n
We can also verify that any SI polynomial with \n\nω\n=\n1\n\n is mapped to a real polynomial through \n\nM\n\nz\n\n\n and any real polynomial is mapped to an SI polynomial with \n\nω\n=\n1\n\n through \n\nW\n\nz\n\n\n. Thus, the set of SI polynomials with \n\nω\n=\n1\n\n is isomorphic to the set of real polynomials. Besides, an SI polynomial with \n\nω\n≠\n1\n\n can be transformed into another one with \n\nω\n=\n1\n\n by performing a suitable uniform rotation of its zeros. It can also be shown that the action of the Möbius transformation over a PSR polynomial leads to a real polynomial that has only even powers. See [4] for more.
\n
\n
\n
4. Zeros location theorems
\n
In this section, we shall discuss some theorems regarding the distribution of the zeros of SC, SR, and SI polynomials on the complex plane. Some general theorems relying on the number of zeros that an arbitrary complex polynomial has inside, on, or outside \n\nS\n\n are also discussed. To save space, we shall not present the proofs of these theorems, which can be found in the original works. Other related theorems can be found in Marden’s book [1].
\n
\n
4.1 Polynomials that do not necessarily have symmetric zeros
\n
The following theorems are classics (see [1] for the proofs):
\n
Theorem 6. (Rouché). Let\n\nq\n\nz\n\n\nand\n\nr\n\nz\n\n\nbe polynomials such that\n\n∣\nq\n\nz\n\n∣\n<\n∣\nr\n\nz\n\n∣\n\nalong all points of\n\nS\n\n. Then, the polynomial\n\np\n\nz\n\n=\nq\n\nz\n\n+\nr\n\nz\n\n\nhas the same number of zeros inside\n\nS\n\nas the polynomial\n\nr\n\nz\n\n\n, counted with multiplicity.
\n
Thus, if a complex polynomial \n\np\n\nz\n\n=\n\np\n0\n\n+\n⋯\n+\n\np\nk\n\n\nz\nk\n\n+\n⋯\n+\n\np\nn\n\n\nz\nn\n\n\n of degree \n\nn\n\n is such that \n\n\n\np\nk\n\n\n>\n\n\n\np\n0\n\n+\n⋯\n+\n\np\n\nk\n−\n1\n\n\n+\n\np\n\nk\n+\n1\n\n\n+\n⋯\n+\n\np\nn\n\n\n\n\n, then \n\np\n\nz\n\n\n will have exactly \n\nk\n\n zeros inside \n\nS\n\n, counted with multiplicity.
\n
Theorem 7. (Gauss and Lucas)The zeros of the derivative\n\n\np\n′\n\n\nz\n\n\nof a polynomial\n\np\n\nz\n\n\nlie all within the convex hull of the zeros of the\n\np\n\nz\n\n\n.
\n
Thereby, if a polynomial \n\np\n\nz\n\n\n has all its zeros on \n\nS\n\n, then all the zeros of \n\n\np\n′\n\n\nz\n\n\n will lie in or on \n\nS\n\n. In particular, the zeros of \n\n\np\n′\n\n\nz\n\n\n will lie on \n\nS\n\n if, and only if, they are multiple zeros of \n\np\n\nz\n\n\n.
\n
Theorem 8. (Cohn)A necessary and sufficient condition for all the zeros of a complex polynomial\n\np\n\nz\n\n\nto lie on\n\nS\n\nis that\n\np\n\nz\n\n\nis SI and that its derivative\n\n\np\n′\n\n\nz\n\n\ndoes not have any zero outside\n\nS\n\n.
\n
Cohn introduced his theorem in [5]. Bonsall and Marden presented a simpler proof of Conh’s theorem in [6] (see also [7]) and applied it to SI polynomials—in fact, this was probably the first paper to use the expression “self-inversive.” Other important result of Cohn is the following: all the zeros of a complex polynomial \n\np\n\nz\n\n=\n\np\nn\n\n\nz\nn\n\n+\n⋯\n+\n\np\n0\n\n\n will lie on \n\nS\n\n if, and only if, \n\n∣\n\np\nn\n\n∣\n=\n∣\n\np\n0\n\n∣\n\n and all the zeros of \n\np\n\nz\n\n\n do not lie outside \n\nS\n\n.
\n
Restricting ourselves to polynomials with real coefficients, Eneström and Kakeya [8, 9, 10] independently presented the following theorem:
\n
Theorem 9. (Eneström and Kakeya)Let\n\np\n\nz\n\n\nbe a polynomial of degree\n\nn\n\nwith real coefficients. If its coefficients are such that\n\n0\n<\n\np\n0\n\n⩽\n\np\n1\n\n⩽\n⋯\n⩽\n\np\n\nn\n−\n1\n\n\n⩽\n\np\nn\n\n\n, then all the zeros of\n\np\n\nz\n\n\nlie in or on\n\nS\n\n. Likewise, if the coefficients of\n\np\n\nz\n\n\nare such that\n\n0\n<\n\np\nn\n\n⩽\n\np\n\nn\n−\n1\n\n\n⩽\n⋯\n⩽\n\np\n1\n\n⩽\n\np\n0\n\n\n, then all the zeros of\n\np\n\nz\n\n\nlie on or outside\n\nS\n\n.
\n
The following theorems are relatively more recent. The distribution of the zeros of a complex polynomial regarding the unit circle \n\nS\n\n was presented by Marden in [1] and slightly enhanced by Jury in [11]:
\n
Theorem 10. (Marden and Jury)Let\n\np\n\nz\n\n\nbe a complex polynomial of degree\n\nn\n\nand\n\n\np\n∗\n\n\nz\n\n\nits reciprocal. Construct the sequence of polynomials\n\n\nP\nj\n\n\nz\n\n=\n\n∑\n\nk\n=\n0\n\n\nn\n−\nj\n\n\n\nP\n\nj\n,\nk\n\n\n\nz\nk\n\n\nsuch that\n\n\nP\n0\n\n\nz\n\n=\np\n\nz\n\n\nand\n\n\nP\n\nj\n+\n1\n\n\n\nz\n\n=\n\n\np\n\nj\n,\n0\n\n\n¯\n\n\nP\nj\n\n\nz\n\n−\n\n\np\n\nj\n,\nn\n−\nj\n\n\n¯\n\n\nP\nj\n∗\n\n\nz\n\n\nfor\n\n0\n⩽\nj\n⩽\nn\n−\n1\n\nso that we have the relations\n\n\np\n\nj\n+\n1\n,\nk\n\n\n=\n\n\np\n\nj\n,\n0\n\n\n¯\n\n\np\n\nj\n,\nk\n\n\n−\n\np\n\nj\n,\nn\n−\nj\n\n\n\n\np\n\nj\n,\nn\n−\nj\n−\nk\n\n\n¯\n\n\n. Let\n\n\nδ\nj\n\n\ndenote the constant terms of the polynomials\n\n\nP\nj\n\n\nz\n\n\n, i.e.,\n\n\nδ\nj\n\n=\n\np\n\nj\n,\n0\n\n\n\nand\n\n\nΔ\nk\n\n=\n\nδ\n1\n\n⋯\n\nδ\nk\n\n\n. Thus, if\n\nN\n\nof the products\n\n\nΔ\nk\n\n\nare negative and\n\nn\n−\nN\n\nof the products\n\n\nΔ\nk\n\n\nare positive so that none of them are zero, then\n\np\n\nz\n\n\nhas\n\nN\n\nzeros inside\n\nS\n\n,\n\nn\n−\nN\n\nzeros outside\n\nS\n\nand no zero on\n\nS\n\n. On the other hand, if\n\n\nΔ\nk\n\n≠\n0\n\nfor some\n\nk\n<\nn\n\nbut\n\n\nP\n\nk\n+\n1\n\n\n\nz\n\n=\n0\n\n, then\n\np\n\nz\n\n\nhas either\n\nn\n−\nk\n\nzeros on\n\nS\n\nor\n\nn\n−\nk\n\nzeros symmetric to\n\nS\n\n. It has additionally\n\nN\n\nzeros inside\n\nS\n\nand\n\nk\n−\nN\n\nzeros outside\n\nS\n\n.
\n
A simple necessary and sufficient condition for all the zeros of a complex polynomial to lie on \n\nS\n\n was presented by Chen in [12]:
\n
Theorem 11. (Chen)A necessary and sufficient condition for all the zeros of a complex polynomial\n\np\n\nz\n\n\nof degree\n\nn\n\nto lie on\n\nS\n\nis that there exists a polynomial\n\nq\n\nz\n\n\nof degree\n\nn\n−\nm\n\nwhose zeros are all in or on\n\nS\n\nand such that\n\np\n\nz\n\n=\n\nz\nm\n\nq\n\nz\n\n+\nω\n\nq\n†\n\n\nz\n\n\nfor some complex number\n\nω\n\nof modulus\n\n1\n\n.
\n
We close this section by mentioning that there exist many other well-known theorems regarding the distribution of the zeros of complex polynomials. We can cite, for example, the famous rule of Descartes (the number of positive zeros of a real polynomial is limited from above by the number of sign variations in the ordered sequence of its coefficients), the Sturm Theorem (the exact number of zeros that a real polynomial has in a given interval \n\n\na\nb\n\n\n of the real line is determined by the formula \n\nN\n=\nvar\n\n\nS\n\nb\n\n\n\n−\nvar\n\n\nS\n\na\n\n\n\n\n, where \n\nvar\n\n\nS\n\nξ\n\n\n\n\n means the number of sign variations of the Sturm sequence \n\nS\n\nx\n\n\n evaluated at \n\nx\n=\nξ\n\n) and Kronecker Theorem (if all the zeros of a monic polynomial with integer coefficients lie on the unit circle, then all these zeros are indeed roots of unity), see [1] for more. There are still other important theorems relying on matrix methods and quadratic forms that were developed by several authors as Cohn, Schur, Hermite, Sylvester, Hurwitz, Krein, among others, see [13].
\n
\n
\n
4.2 Real self-reciprocal polynomials
\n
Let us now consider real SR polynomials. The theorems below are usually applied to PSR polynomials, but some of them can be extended to NSR polynomials as well.
\n
An analog of Eneström-Kakeya theorem for PSR polynomials was found by Chen in [12] and then, in a slightly stronger version, by Chinen in [14]:
\n
Theorem 12. (Chen and Chinen)Let\n\np\n\nz\n\n\nbe a PSR polynomial of degree\n\nn\n\nthat is written in the form\n\np\n\nz\n\n=\n\np\n0\n\n+\n\np\n1\n\nz\n+\n⋯\n+\n\np\nk\n\n\nz\nk\n\n+\n\np\nk\n\n\nz\n\nn\n−\nk\n\n\n+\n\np\n\nk\n−\n1\n\n\n\nz\n\nn\n−\nk\n+\n1\n\n\n+\n⋯\n+\n\np\n0\n\n\nz\nn\n\n\nand such that\n\n0\n<\n\np\nk\n\n<\n\np\n\nk\n−\n1\n\n\n<\n⋯\n<\n\np\n1\n\n<\n\np\n0\n\n\n. Then all the zeros of\n\np\n\nz\n\n\nare on\n\nS\n\n.
\n
Going in the same direction, Choo found in [15] the following condition:
\n
Theorem 13. (Choo)Let\n\np\n\nz\n\n\nbe a PSR polynomial of degree\n\nn\n\nand such that its coefficients satisfy the following conditions:\n\n\nnp\nn\n\n⩾\n\n\nn\n−\n1\n\n\n\np\n\nn\n−\n1\n\n\n⩾\n⋯\n⩾\n\n\nk\n+\n1\n\n\n\np\n\nk\n+\n1\n\n\n>\n0\n\nand\n\n\n\nk\n+\n1\n\n\n\np\n\nk\n+\n1\n\n\n\n⩾\n\n\n∑\n\nj\n=\n0\n\nk\n\n\n\n\n\nj\n+\n1\n\n\n\np\n\nj\n+\n1\n\n\n−\n\njp\nj\n\n\n\n\nfor\n\n0\n⩽\nk\n⩽\nn\n−\n1\n\n. Then, all the zeros of\n\np\n\nz\n\n\nare on\n\nS\n\n.
\n
Lakatos discussed the separation of the zeros on the unit circle of PSR polynomials in [16]; she also found several sufficient conditions for their zeros to be all on \n\nS\n\n. One of the main theorems is the following:
\n
Theorem 14. (Lakatos)Let\n\np\n\nz\n\n\nbe a PSR polynomial of degree\n\nn\n>\n2\n\n. If\n\n\n\np\nn\n\n\n⩾\n\n∑\n\nk\n=\n1\n\n\nn\n−\n1\n\n\n\n\n\np\nn\n\n−\n\np\nk\n\n\n\n\n, then all the zeros of\n\np\n\nz\n\n\nlie on\n\nS\n\n. Moreover, the zeros of\n\np\n\nz\n\n\nare all simple, except when the equality takes place.
\n
For PSR polynomials of odd degree, Lakatos and Losonczi [17] found a stronger version of this result:
\n
Theorem 15. (Lakatos and Losonczi)Let\n\np\n\nz\n\n\nbe a PSR polynomial of odd degree, say\n\nn\n=\n2\nm\n+\n1\n\n. If\n\n\n\np\n\n2\nm\n+\n1\n\n\n\n⩾\n\ncos\n2\n\n\n\nϕ\nm\n\n\n\n∑\n\nk\n=\n1\n\n\n2\nm\n\n\n\n\n\np\n\n2\nm\n+\n1\n\n\n−\n\np\nk\n\n\n\n\n, where\n\n\nϕ\nm\n\n=\nπ\n/\n\n\n4\n\n\nm\n+\n1\n\n\n\n\n\n, then all the zeros of\n\np\n\nz\n\n\nlie on\n\nS\n\n. The zeros are simple except when the equality is strict.
\n
Theorem 14 was generalized further by Lakatos and Losonczi in [18]:
\n
Theorem 16. (Lakatos and Losonczi)All zeros of a PSR polynomial\n\np\n\nz\n\n\nof degree\n\nn\n>\n2\n\nlie on\n\nS\n\nif the following conditions hold:\n\n\n\n\np\nn\n\n+\nr\n\n\n⩾\n\n∑\n\nk\n=\n1\n\n\nn\n−\n1\n\n\n\n\n\np\nk\n\n−\n\np\nn\n\n+\nr\n\n\n\n,\n\n\np\nn\n\nr\n⩾\n0\n\n, and\n\n\n\np\nn\n\n\n⩾\n\nr\n\n\n, for\n\nr\n∈\nR\n\n.
\n
Other conditions for all the zeros of a PSR polynomial to lie on \n\nS\n\n were presented by Kwon in [19]. In its simplest form, Kown’s theorem can be enunciated as follows:
\n
Theorem 17. (Kwon)Let\n\np\n\nz\n\n\nbe a PSR polynomial of even degree\n\nn\n⩾\n2\n\nwhose leading coefficient\n\n\np\nn\n\n\nis positive and\n\n\np\n0\n\n⩽\n\np\n1\n\n⩽\n⋯\n⩽\n\np\nn\n\n\n. In this case, all the zeros of\n\np\n\nz\n\n\nwill lie on\n\nS\n\nif, either\n\n\np\n\nn\n/\n2\n\n\n⩾\n\n∑\n\nk\n=\n0\n\nn\n\n\n\n\np\nk\n\n−\n\np\n\nn\n/\n2\n\n\n\n\n\n, or\n\np\n\n1\n\n⩾\n0\n\nand\n\n\np\nn\n\n⩾\n\n1\n2\n\n\n∑\n\nk\n=\n1\n\n\nn\n−\n1\n\n\n\n\n\np\nk\n\n−\n\np\n\nn\n/\n2\n\n\n\n\n\n.
\n
Modified forms of this theorem hold for PSR polynomials of odd degree and for the case where the coefficients of \n\np\n\nz\n\n\n do not have the ordination above—see [19] for these cases. Kwon also found conditions for all but two zeros of \n\np\n\nz\n\n\n to lie on \n\nS\n\n in [20], which is relevant to the theory of Salem polynomials—see Section 5.
\n
Other interesting results are the following: Konvalina and Matache [21] found conditions under which a PSR polynomial has at least one non-real zero on \n\nS\n\n. Kim and Park [22] and then Kim and Lee [23] presented conditions for which all the zeros of certain PSR polynomials lie on \n\nS\n\n (some open cases were also addressed by Botta et al. in [24]). Suzuki [25] presented necessary and sufficient conditions, relying on matrix algebra and differential equations, for all the zeros of PSR polynomials to lie on \n\nS\n\n. In [26] Botta et al. studied the distribution of the zeros of PSR polynomials with a small perturbation in their coefficients. Real SR polynomials of height \n\n1\n\n—namely, special cases of Littlewood, Newman, and Borwein polynomials—were studied by several authors, see [27, 28, 29, 30, 31, 32, 33, 34, 35] and references therein.2 Zeros of the so-called Ramanujan Polynomials and generalizations were analyzed in [37, 38, 39]. Finally, the Galois theory of PSR polynomials was studied in [40] by Lindstrøm, who showed that any PSR polynomial of degree less than \n\n10\n\n can be solved by radicals.
\n
\n
\n
4.3 Complex self-reciprocal and self-inversive polynomials
\n
Let us consider now the case of complex SR polynomials and SI polynomials. Here, we remark that many of the theorems that hold for SI polynomials either also hold for SR polynomials or can be easily adapted to this case (the opposite is also true).
\n
Theorem 18. (Cohn)An SI polynomial\n\np\n\nz\n\n\nhas as many zeros outside\n\nS\n\nas does its derivative\n\n\np\n′\n\n\nz\n\n\n.
\n
This follows directly from Cohn’s Theorem 8 for the case where \n\np\n\nz\n\n\n is SI. Besides, we can also conclude from this that the derivative of \n\np\n\nz\n\n\n has no zeros on \n\nS\n\n except at the multiple zeros of \n\np\n\nz\n\n\n. Furthermore, if an SI polynomial \n\np\n\nz\n\n\n of degree \n\nn\n\n has exactly \n\nk\n\n zeros on \n\nS\n\n, while its derivative has exactly \n\nl\n\n zeros in or on \n\nS\n\n, both counted with multiplicity, then \n\nn\n=\n2\n\n\nl\n+\n1\n\n\n−\nk\n\n.
\n
O’Hara and Rodriguez [41] showed that the following conditions are always satisfied by SI polynomials whose zeros are all on \n\nS\n\n:
\n
Theorem 19. (O’Hara and Rodriguez)Let\n\np\n\nz\n\n\nbe an SI polynomial of degree\n\nn\n\nwhose zeros are all on\n\nS\n\n. Then, the following inequality holds:\n\n\n∑\n\nj\n=\n0\n\nn\n\n\n\n\np\nj\n\n\n2\n\n⩽\n\n\n\np\n\nz\n\n\n\n2\n\n\n, where\n\n\n\np\n\nz\n\n\n\n\ndenotes the maximum modulus of\n\np\n\nz\n\n\non the unit circle; besides, if this inequality is strict then the zeros of\n\np\n\nz\n\n\nare rotations of\n\nn\n\nth roots of unity. Moreover, the following inequalities are also satisfied:\n\n\n\na\nk\n\n\n⩽\n\n1\n2\n\n\n\np\n\nz\n\n\n\n\n if \n\nk\n≠\nn\n/\n2\n\n and \n\n\n\na\nk\n\n\n⩽\n\n\n2\n\n2\n\n\n\np\n\nz\n\n\n\n\n for \n\nk\n=\nn\n/\n2\n\n.
\n
Schinzel in [42], generalized Lakatos Theorem 14 for SI polynomials:
\n
Theorem 20. (Schinzel)Let\n\np\n\nz\n\n\nbe an SI polynomial of degree\n\nn\n\n. If the inequality\n\n\n\np\nn\n\n\n⩾\n\ninf\n\na\n,\nb\n∈\nC\n:\n∣\nb\n∣\n=\n1\n\n\n\n∑\n\nk\n=\n0\n\nn\n\n\n\n\nap\nk\n\n−\n\nb\n\nn\n−\nk\n\n\n\np\nn\n\n\n\n\n, then all the zeros of\n\np\n\nz\n\n\nlie on\n\nS\n\n. These zero are simple whenever the equality is strict.
\n
In a similar way, Losonczi and Schinzel [43] generalized theorem 15 for the SI case:
\n
Theorem 21. (Losonczi and Schinzel)Let\n\np\n\nz\n\n\nbe an SI polynomial of odd degree, i.e.,\n\nn\n=\n2\nm\n+\n1\n\n. If\n\n\n\np\n\n2\nm\n+\n1\n\n\n\n⩾\n\ncos\n2\n\n\n\nϕ\nm\n\n\n\ninf\n\na\n,\nb\n∈\nC\n:\n∣\nb\n∣\n=\n1\n\n\n\n∑\n\nk\n=\n1\n\n\n2\nm\n+\n1\n\n\n\n\n\nap\nk\n\n−\n\nb\n\n2\nm\n+\n1\n−\nk\n\n\n\np\n\n2\nm\n+\n1\n\n\n\n\n\n, where\n\n\nϕ\nm\n\n=\nπ\n/\n\n\n4\n\n\nm\n+\n1\n\n\n\n\n\n, then all the zeros of\n\np\n\nz\n\n\nlie on\n\nS\n\n. The zeros are simple except when the equality is strict.
\n
Another sufficient condition for all the zeros of an SI polynomial to lie on \n\nS\n\n was presented by Lakatos and Losonczi in [44]:
\n
Theorem 22. (Lakatos and Losonczi)Let\n\np\n\nz\n\n\nbe an SI polynomial of degree\n\nn\n\nand suppose that the inequality\n\n\n\np\nn\n\n\n⩾\n\n\n1\n2\n\n\n∑\n\nk\n=\n1\n\n\nn\n−\n1\n\n\n\n\np\nk\n\n\n\nholds. Then, all the zeros of\n\np\n\nz\n\n\nlie on\n\nS\n\n. Moreover, the zeros are all simple except when an equality takes place.
\n
In [45], Lakatos and Losonczi also formulated a theorem that contains as special cases many of the previous results:
\n
Theorem 23. (Lakatos and Losonczi)Let\n\np\n\nz\n\n=\n\np\n0\n\n+\n⋯\n+\n\np\nn\n\n\nz\nn\n\n\nbe an SI polynomial of degree\n\nn\n⩾\n2\n\nand\n\na\n\n,\n\nb\n\n, and\n\nc\n\nbe complex numbers such that\n\na\n≠\n0\n\n,\n\n∣\nb\n∣\n=\n1\n\n, and\n\nc\n/\n\np\nn\n\n∈\nR\n\n,\n\n0\n⩽\nc\n/\n\np\nn\n\n⩽\n1\n\n. If\n\n\n\n\np\nn\n\n+\nc\n\n\n⩾\n\n\n\nap\n0\n\n−\n\nb\nn\n\n\np\nn\n\n\n\n+\n\n∑\n\nk\n=\n1\n\n\nn\n−\n1\n\n\n\n\n\nap\nk\n\n−\n\nb\n\nn\n−\nk\n\n\n\n\nc\n−\n\np\nn\n\n\n\n\n\n+\n\n\n\nap\nn\n\n−\n\np\nn\n\n\n\n\n, then, all the zeros of\n\np\n\nz\n\n\nlie on\n\nS\n\n. Moreover, these zeros are simple if the inequality is strict.
\n
In [46], Losonczi presented the following necessary and sufficient conditions for all the zeros of a (complex) SR polynomial of even degree to lie on \n\nS\n\n:
\n
Theorem 24. (Losonczi)Let\n\np\n\nz\n\n\nbe a monic complex SR polynomial of even degree, say\n\nn\n=\n2\nm\n\n. Then, all the zeros of\n\np\n\nz\n\n\nwill lie on\n\nS\n\nif, and only if, there exist real numbers\n\n\nα\n1\n\n,\n…\n,\n\nα\n\n2\nm\n\n\n\n, all with moduli less than or equal to\n\n2\n\n, that satisfy the inequalities:\n\n\np\nk\n\n=\n\n\n\n−\n1\n\n\nk\n\n\n∑\n\nl\n=\n0\n\n\n\nk\n/\n2\n\n\n\n\n\n\n\nm\n−\nk\n+\n2\nl\n\n\n\n\nl\n\n\n\n\n\nσ\n\nk\n−\n2\nl\n\n\n2\nm\n\n\n\n\nα\n1\n\n…\n\nα\n\n2\nm\n\n\n\n\n,\n\n0\n⩽\nk\n⩽\nm\n\n, where\n\n\nσ\nk\n\n2\nm\n\n\n\n\nα\n1\n\n…\n\nα\n\n2\nm\n\n\n\n\ndenotes the\n\nk\n\nth elementary symmetric function in the\n\n2\nm\n\nvariables\n\n\nα\n1\n\n,\n…\n,\n\nα\n\n2\nm\n\n\n\n.
\n
Losonczi, in [46], also showed that if all the zeros of a complex monic reciprocal polynomial are on \n\nS\n\n, then its coefficients are all real and satisfy the inequality \n\n\n\np\nn\n\n\n⩽\n\n\n\n\nn\n\n\n\n\nk\n\n\n\n\n\n for \n\n0\n⩽\nk\n⩽\nn\n\n.
\n
The theorems above give conditions for all the zeros of SI or SR polynomials to lie on \n\nS\n\n. In many cases, however, we need to verify if a polynomial has a given number of zeros (or none) on the unit circle. Considering this problem, Vieira in [47] found sufficient conditions for an SI polynomial of degree \n\nn\n\n to have a determined number of zeros on the unit circle. In terms of the length, \n\nL\n\n\np\n\nz\n\n\n\n=\n\n\np\n0\n\n\n+\n⋯\n+\n\n\np\nn\n\n\n\n of a polynomial \n\np\n\nz\n\n\n of degree \n\nn\n\n, this theorem can be stated as follows:
\n
Theorem 25. (Vieira)Let\n\np\n\nz\n\n\nbe an SI polynomial of degree\n\nn\n\n. If the inequality\n\n\n\np\n\nn\n−\nm\n\n\n\n⩾\n\n1\n4\n\n\n\nn\n\nn\n−\nm\n\n\n\nL\n\n\np\n\nz\n\n\n\n\n,\n\nm\n<\nn\n/\n2\n\n, holds true, then\n\np\n\nz\n\n\nwill have exactly\n\nn\n−\n2\nm\n\nzeros on\n\nS\n\n; besides, all these zeros are simple when the inequality is strict. Moreover,\n\np\n\nz\n\n\nwill have no zero on\n\nS\n\nif, for\n\nn\n\neven and\n\nm\n=\nn\n/\n2\n\n, the inequality\n\n\n\np\nm\n\n\n>\n\n1\n2\n\nL\n\n\np\n\nz\n\n\n\n\nis satisfied.
\n
The case \n\nm\n=\n0\n\n corresponds to Lakatos and Losonczi Theorem 14 for all the zeros of \n\np\n\nz\n\n\n to lie on \n\nS\n\n. The necessary counterpart of this theorem was considered by Stankov in [48], with an application to the theory of Salem numbers—see Section 5.1.
\n
Other results on the distribution of zeros of SI polynomials include the following: Sinclair and Vaaler [49] showed that a monic SI polynomial \n\np\n\nz\n\n\n of degree \n\nn\n\n satisfying the inequalities \n\n\nL\nr\n\n\n\np\n\nz\n\n\n\n⩽\n2\n+\n\n2\nr\n\n\n\n\nn\n−\n1\n\n\n\n1\n−\nr\n\n\n\n or \n\n\nL\nr\n\n\n\np\n\nz\n\n\n\n⩽\n2\n+\n\n2\nr\n\n\n\n\nl\n−\n2\n\n\n\n1\n−\nr\n\n\n\n, where \n\nr\n⩾\n1\n\n, \n\n\nL\nr\n\n\n\np\n\nz\n\n\n\n=\n\n\n\np\n0\n\n\nr\n\n+\n⋯\n+\n\n\n\np\nn\n\n\nr\n\n\n, and \n\nl\n\n is the number of non-null terms of \n\np\n\nz\n\n\n, has all their zeros on \n\nS\n\n; the authors also studied the geometry of SI polynomials whose zeros are all on \n\nS\n\n. Choo and Kim applied Theorem 11 to SI polynomials in [50]. Hypergeometric polynomials with all their zeros on \n\nS\n\n were considered in [51, 52]. Kim [53] also obtained SI polynomials which are related to Jacobi polynomials. Ito and Wimmer [54] studied SI polynomial operators in Hilbert space whose spectrum is on \n\nS\n\n.
\n
\n
\n
\n
5. Where these polynomials are found?
\n
In this section, we shall briefly discuss some important or recent applications of the theory of polynomials with symmetric zeros. We remark, however, that our selection is by no means exhaustive: for example, SR and SI polynomials also find applications in many fields of mathematics (e.g., information and coding theory [55], algebraic curves over a finite field and cryptography [56], elliptic functions [57], number theory [58], etc.) and physics (e.g., Lee-Yang theorem in statistical physics [59], Poincaré Polynomials defined on Calabi-Yau manifolds of superstring theory [60], etc.).
\n
\n
5.1 Polynomials with small Mahler measure
\n
Given a monic polynomial \n\np\n\nz\n\n\n of degree \n\nn\n\n, with integer coefficients, the Mahler measure of \n\np\n\nz\n\n\n, denoted by \n\nM\n\n\np\n\nz\n\n\n\n\n, is defined as the product of the modulus of all those zeros of \n\np\n\nz\n\n\n that lie in the exterior of \n\nS\n\n [61]. That is
where \n\n\nζ\n1\n\n,\n…\n,\n\nζ\nn\n\n\n are the zeros3 of \n\np\n\nz\n\n\n. Thus, if a monic integer polynomial \n\np\n\nz\n\n\n has all its zeros in or on the unit circle, we have \n\nM\n\n\np\n\nz\n\n\n\n=\n1\n\n; in particular, all cyclotomic polynomials (which are PSR polynomials whose zeros are the primitive roots of unity, see [1]) have Mahler measure equal to \n\n1\n\n. In a sense, the Mahler measure of a polynomial \n\np\n\nz\n\n\n measures how close it is to the cyclotomic polynomials. Therefore, it is natural to raise the following:
\n
Problem 1. (Mahler)Find the monic, integer, non-cyclotomic polynomial with the smallest Mahler measure.
\n
This is an 80-year-old open problem of mathematics. Of course, we can expect that the polynomials with the smallest Mahler measure be among those with only a few number of zeros outside \n\nS\n\n, in particular among those with only one zero outside \n\nS\n\n. A monic integer polynomial that has exactly one zero outside \n\nS\n\n is called a Pisot polynomial and its unique zero of modulus greater than \n\n1\n\n is called its Pisot number [62]. A breakthrough towards the solution of Mahler’s problem was given by Smyth in [63]:
\n
Theorem 26. (Smyth)The Pisot polynomial\n\nS\n\nz\n\n=\n\nz\n3\n\n−\nz\n−\n1\n\nis the polynomial with smallest Mahler measure among the set of all monic, integer, and non-SR polynomials. Its Mahler measure is given by the value of its Pisot number, which is,
The Mahler problem is, however, still open for SR polynomials. A monic integer SR polynomial with exactly two (real and positive) zeros (say, \n\nζ\n\n and \n\n1\n/\nζ\n\n) not lying on \n\nS\n\n is called a Salem polynomial [62, 64]. It can be shown that a Pisot polynomial with at least one zero on \n\nS\n\n is also a Salem polynomial. The unique positive zero greater than one of a Salem polynomial is called its Salem number, which also equals the value of its Mahler measure. A Salem number \n\ns\n\n is said to be small if \n\ns\n<\nσ\n\n; up to date, only \n\n47\n\n small Salem numbers are known [65, 66] and the smallest known one was found about 80 years ago by Lehmer [67]. This gave place to the following:
\n
Conjecture 1. (Lehmer)The monic integer polynomial with the smallest Mahler measure is the Lehmer polynomial\n\nL\n\nz\n\n=\n\nz\n10\n\n+\n\nz\n9\n\n−\n\nz\n7\n\n−\n\nz\n6\n\n−\n\nz\n5\n\n−\n\nz\n4\n\n−\n\nz\n3\n\n+\nz\n+\n1\n\n, a Salem polynomial whose Mahler measure is\n\nΛ\n≈\n1.17628081826\n\n, known as Lehmer’s constant.
\n
The proof of this conjecture is also an open problem. To be fair, we do not even know if there exists a smallest Salem number at all. This is the content of another problem raised by Lehmer:
\n
Problem 2. (Lehmer)Answer whether there exists or not a positive number\n\nϵ\n\nsuch that the Mahler measure of any monic, integer, and non-cyclotomic polynomial\n\np\n\nz\n\n\nsatisfies the inequality\n\nM\n\n\np\n\nz\n\n\n\n>\n1\n+\nϵ\n\n.
\n
Lehmer’s polynomial also appears in connection with several fields of mathematics. Many examples are discussed in Hironaka’s paper [68]; here we shall only present an amazing identity found by Bailey and Broadhurst in [69] in their works on polylogarithm ladders: if \n\nλ\n\n is any zero of the aforementioned Lehmer’s polynomial \n\nL\n\nz\n\n\n, then,
A knot is a closed, non-intersecting, one-dimensional curve embedded on \n\n\nR\n3\n\n\n [70]. Knot theory studies topological properties of knots as, for example, criteria under which a knot can be unknot, conditions for the equivalency between knots, the classification of prime knots, etc.; see [70] for the corresponding definitions. In Figure 1, we plotted all prime knots up to six crossings.
\n
Figure 1.
A table of prime knots up to six crossings. In the Alexander-Briggs notation these knots are, in order, \n\n\n0\n1\n\n\n, \n\n\n3\n1\n\n\n, \n\n\n4\n1\n\n\n, \n\n\n5\n1\n\n\n, \n\n\n5\n2\n\n\n, \n\n\n6\n1\n\n\n, \n\n\n6\n2\n\n\n, and \n\n\n6\n3\n\n\n.
\n
One of the most important questions in knot theory is to determine whether or not two knots are equivalent. This, however, is not an easy task. A way of attacking this question is to look for abstract objects—mainly the so-called knot invariants—rather than to the knots themselves. A knot invariant is a (topologic, combinatorial, algebraic, etc.) quantity that can be computed for any knot and that is always the same for equivalents knots.4 An important class of knot invariants is constituted by the so-called Knot Polynomials. Knot polynomials were introduced in 1928 by Alexander [71]. They consist in polynomials with integer coefficients that can be written down for every knot. For about 60 years since its creation, Alexander polynomials were the only known kind of knot polynomial. It was only in 1985 that Jones [72] came up with a new kind of knot polynomials—today known as Jones polynomials—and since then other kinds were discovered as well, see [70].
\n
What is interesting for us here is that the Alexander polynomials are PSR polynomials of even degree (say, \n\nn\n=\n2\nm\n\n) and with integer coefficients.5 Thus, they have the following general form:
where \n\n\nδ\ni\n\n∈\nN\n\n, \n\n0\n⩽\ni\n⩽\nm\n\n. In Table 1, we present the $\\delta_{m - 1}$Alexander polynomials for the prime knots up to six crossings.
Alexander polynomials for prime knots up to six crossings.
\n
Knots theory finds applications in many fields of mathematics in physics—see [70]. In mathematics, we can cite a very interesting connection between Alexander polynomials and the theory of Salem numbers: more precisely, the Alexander polynomial associated with the so-called Pretzel Knot\n\nP\n\n\n−\n2,3,7\n\n\n\n is nothing but the Lehmer polynomial \n\nL\n\nz\n\n\n introduced in Section 5.1; it is indeed the Alexander polynomial with the smallest Mahler measure [73]. In physics, knot theory is connected with quantum groups and it also can be used to one construct solutions of the Yang-Baxter equation [74] through a method called baxterization of braid groups.
\n
\n
\n
5.3 Bethe equations
\n
Bethe equations were introduced in 1931 by Hans Bethe [75], together with his powerful method—the so-called Bethe Ansatz Method—for solving spectral problems associated with exactly integrable models of statistical mechanics. They consist in a system of coupled and non-linear equations that ensure the consistency of the Bethe Ansatz. In fact, for the XXZ Heisenberg spin chain, the Bethe Equations consist in a coupled system of trigonometric equations; however, after a change of variables is performed, we can write them in the following rational form:
where L ∈ \n\nN\n\n is the length of the chain, N ∈ \n\nN\n\n is the excitation number and Δ ∈ \n\nR\n\n is the so-called spectral parameter. A solution of (18) consists in a (non-ordered) set \n\nX\n=\n\n\nx\n1\n\n…\n\nx\nN\n\n\n\n of the unknowns \n\n\nx\n1\n\n,\n…\n,\n\nx\nN\n\n\n so that (18) is satisfied. Notice that the Bethe equations satisfy the important relation \n\n\nx\n1\nL\n\n\nx\n2\nL\n\n⋯\n\nx\nN\nL\n\n=\n1\n\n, which suggests an inversive symmetry of their zeros.
\n
In [76], Vieira and Lima-Santos showed that the solutions of (18), for \n\nN\n=\n2\n\n and arbitrary \n\nL\n\n, are given in terms of the zeros of certain SI polynomials. In fact, (18) becomes a system of two coupled algebraic equations for \n\nN\n=\n2\n\n, namely,
Now, from the relation \n\n\nx\n1\nL\n\n\nx\n2\nL\n\n=\n1\n\n we can eliminate one of the unknowns in (19)—for instance, by setting \n\n\nx\n2\n\n=\n\nω\na\n\n/\n\nx\n1\n\n\n, where \n\n\nω\na\n\n=\nexp\n\n\n2\nπia\n/\nL\n\n\n\n, \n\n1\n⩽\na\n⩽\nL\n\n, are the roots of unity of degree \n\nL\n\n. Replacing these values for \n\n\nx\n2\n\n\n into (19), we obtain the following polynomial equations fixing \n\n\nx\n1\n\n\n:
We can easily verify that the polynomial \n\n\np\na\n\n\nz\n\n\n is SI for each value of \n\na\n\n. They also satisfy the relations \n\n\np\na\n\n\nz\n\n=\n\nz\nL\n\np\n\n\n\nω\na\n\n/\nz\n\n\n\n, \n\n1\n⩽\na\n⩽\nL\n\n, which means that the solutions of (19) have the general form \n\nX\n=\n\nζ\n\n\nω\na\n\n/\nζ\n\n\n\n for \n\nζ\n\n any zero of \n\n\np\na\n\n\nz\n\n\n. In [76], the distribution of the zeros of the polynomials \n\n\np\na\n\n\nz\n\n\n was analyzed through an application of Vieira’s Theorem 25. It was shown that the exact behavior of the zeros of the polynomials \n\n\np\na\n\n\nz\n\n\n, for each \n\na\n\n, depends on two critical values of \n\nΔ\n\n, namely
as follows: if \n\n∣\nΔ\n∣\n⩽\n\nΔ\na\n\n1\n\n\n\n, then all the zeros of \n\n\np\na\n\n\nz\n\n\n are on \n\nS\n\n; if \n\n∣\nΔ\n∣\n⩾\n\nΔ\na\n\n2\n\n\n\n, then all the zeros of \n\n\np\na\n\n\nz\n\n\n but two are on \n\nS\n\n; (see [76] for the case \n\n\nΔ\na\n\n1\n\n\n<\n∣\nΔ\n∣\n<\n\nΔ\na\n\n2\n\n\n\n and more details).
\n
Finally, we highlight that the polynomial \n\n\np\na\n\n\nz\n\n\n becomes a Salem polynomial for \n\na\n=\nL\n\n and integer values of \n\nΔ\n\n. This was one of the first appearances of Salem polynomials in physics.
\n
\n
\n
5.4 Orthogonal polynomials
\n
An infinite sequence \n\nP\n=\n\n\n\n\nP\nn\n\n\nz\n\n\n\n\nn\n∈\nN\n\n\n\n of polynomials \n\n\nP\nn\n\n\nz\n\n\n of degree \n\nn\n\n is said to be an orthogonal polynomial sequence on the interval \n\n\nl\nr\n\n\n of the real line if there exists a function \n\nw\n\nx\n\n\n, positive in \n\n\nl\nr\n\n∈\nR\n\n, such that
where \n\n\nK\n0\n\n\n, \n\n\nK\n1\n\n\n, etc. are positive numbers. Orthogonal polynomial sequences on the real line have many interesting and important properties—see [77].
\n
Very recently, Vieira and Botta [78, 79] studied the action of Möbius transformations over orthogonal polynomial sequences on the real line. In particular, they showed that the infinite sequence \n\nT\n=\n\n\n\n\nT\nn\n\n\nz\n\n\n\n\nn\n∈\nN\n\n\n\n of the Möbius-transformed polynomials \n\n\nT\nn\n\n\nz\n\n=\n\n\n\nz\n−\n1\n\n\nn\n\n\nP\nn\n\n\n\nW\n\nz\n\n\n\n\n, where \n\nW\n\nz\n\n=\n−\ni\n\n\nz\n+\n1\n\n\n/\n\n\nz\n−\n1\n\n\n\n, is an SI polynomial sequence with all their zeros on the unit circle \n\nS\n\n—see Table 2 for an example. We highlight that the polynomials \n\n\nT\nn\n\n\nz\n\n∈\nT\n\n also have properties similar to the original polynomials \n\n\nP\nn\n\n\nz\n\n∈\nP\n\n as, for instance, they satisfy an orthogonality condition on the unit circle and a three-term recurrence relation, their zeros lie all on \n\nS\n\n and are simple, for \n\nn\n⩾\n1\n\n the zeros of \n\n\nT\nn\n\n\nz\n\n\n interlaces with those of \n\n\nT\n\nn\n+\n1\n\n\n\nz\n\n\n and so on—see [78, 79] for more details.
Hermite and Möbius-transformed Hermite polynomials, up to \n\n4\n\nth degree.
\n
\n
\n
\n
6. Conclusions
\n
In this work, we reviewed the theory of self-conjugate, self-reciprocal, and self-inversive polynomials. We discussed their main properties, how they are related to each other, the main theorems regarding the distribution of their zeros and some applications of these polynomials both in physics and mathematics. We hope that this short review suits for a compact introduction of the subject, paving the way for further developments in this interesting field of research.
\n
\n
Acknowledgments
\n
We thank the editorial staff for all the support during the publishing process and also the Coordination for the Improvement of Higher Education (CAPES).
\n
\n',keywords:"self-inversive polynomials, self-reciprocal polynomials, Pisot and Salem polynomials, Möbius transformations, knot theory, Bethe equations",chapterPDFUrl:"https://cdn.intechopen.com/pdfs/66332.pdf",chapterXML:"https://mts.intechopen.com/source/xml/66332.xml",downloadPdfUrl:"/chapter/pdf-download/66332",previewPdfUrl:"/chapter/pdf-preview/66332",totalDownloads:250,totalViews:163,totalCrossrefCites:0,totalDimensionsCites:0,hasAltmetrics:1,dateSubmitted:"August 11th 2018",dateReviewed:"November 26th 2018",datePrePublished:"March 23rd 2019",datePublished:null,readingETA:"0",abstract:"Polynomials whose zeros are symmetric either to the real line or to the unit circle are very important in mathematics and physics. We can classify them into three main classes: the self-conjugate polynomials, whose zeros are symmetric to the real line; the self-inversive polynomials, whose zeros are symmetric to the unit circle; and the self-reciprocal polynomials, whose zeros are symmetric by an inversion with respect to the unit circle followed by a reflection in the real line. Real self-reciprocal polynomials are simultaneously self-conjugate and self-inversive so that their zeros are symmetric to both the real line and the unit circle. In this survey, we present a short review of these polynomials, focusing on the distribution of their zeros.",reviewType:"peer-reviewed",bibtexUrl:"/chapter/bibtex/66332",risUrl:"/chapter/ris/66332",book:{slug:"polynomials-theory-and-application"},signatures:"Ricardo Vieira",authors:null,sections:[{id:"sec_1",title:"1. Introduction",level:"1"},{id:"sec_2",title:"2. Self-conjugate, self-reciprocal, and self-inversive polynomials",level:"1"},{id:"sec_3",title:"3. How these polynomials are related to each other?",level:"1"},{id:"sec_4",title:"4. Zeros location theorems",level:"1"},{id:"sec_4_2",title:"4.1 Polynomials that do not necessarily have symmetric zeros",level:"2"},{id:"sec_5_2",title:"4.2 Real self-reciprocal polynomials",level:"2"},{id:"sec_6_2",title:"4.3 Complex self-reciprocal and self-inversive polynomials",level:"2"},{id:"sec_8",title:"5. Where these polynomials are found?",level:"1"},{id:"sec_8_2",title:"5.1 Polynomials with small Mahler measure",level:"2"},{id:"sec_9_2",title:"5.2 Knot theory",level:"2"},{id:"sec_10_2",title:"5.3 Bethe equations",level:"2"},{id:"sec_11_2",title:"5.4 Orthogonal polynomials",level:"2"},{id:"sec_13",title:"6. Conclusions",level:"1"},{id:"sec_14",title:"Acknowledgments",level:"1"}],chapterReferences:[{id:"B1",body:'Marden M. Geometry of Polynomials. 2nd ed. Vol. 3. Providence, Rhode Island: American Mathematical Society; 1966\n'},{id:"B2",body:'Milovanović GV, Mitrinović DS, Rassias TM. Topics in Polynomials: Extremal Problems, Inequalities, Zeros. Singapore: World Scientific; 1994\n'},{id:"B3",body:'Sheil-Small T. Complex Polynomials. Vol. 75. Cambridge: Cambridge University Press; 2002\n'},{id:"B4",body:'Vieira R. How to count the number of zeros that a polynomial has on the unit circle? 2019. arXiv preprint: arXiv:1902.04231\n'},{id:"B5",body:'Cohn A. Über die Anzahl der Wurzeln einer algebraischen Gleichung in einem Kreise. Mathematische Zeitschrift. 1922;14(1):110-148. DOI: 10.1007/BF01215894\n'},{id:"B6",body:'Bonsall F, Marden M. Zeros of self-inversive polynomials. Proceedings of the American Mathematical Society. 1952;3(3):471-475. DOI: 10.2307/2031905\n'},{id:"B7",body:'Ancochea G. Zeros of self-inversive polynomials. Proceedings of the American Mathematical Society. 1953;4(6):900-902. DOI: 10.2307/2031826\n'},{id:"B8",body:'Eneström G. Härledning af en allmän formel för antalet pensionärer som vid en godtycklig tidpunkt förefinnas inom en sluten pensionskassa. Öfversigt af Kongl. Vetenskaps-Akademiens Förhandlingar. 1893;50:405-415\n'},{id:"B9",body:'Eneström G. Remarque sur un théorème relatif aux racines de l’équation \n\n\na\nn\n\n\nx\nn\n\n+\n\na\n\nn\n−\n1\n\n\n\nx\n\nn\n−\n1\n\n\n+\n⋯\n+\n\na\n1\n\nx\n+\n\na\n0\n\n=\n0\n\n où tous les coefficientes a sont réels et positifs. Tohoku Mathematical Journal, First Series. 1920;18:34-36\n'},{id:"B10",body:'Kakeya S. On the limits of the roots of an algebraic equation with positive coefficients. Tohoku Mathematical Journal, First Series. 1912;2:140-142\n'},{id:"B11",body:'Jury E. A note on the reciprocal zeros of a real polynomial with respect to the unit circle. IEEE Transactions on Circuit Theory. 1964;11(2):292-294. DOI: 10.1109/TCT.1964.1082289\n'},{id:"B12",body:'Chen W. On the polynomials with all their zeros on the unit circle. Journal of Mathematical Analysis and Applications. 1995;190(3):714-724. DOI: 10.1006/jmaa.1995.1105\n'},{id:"B13",body:'Krein M, Naimark M. The method of symmetric and Hermitian forms in the theory of the separation of the roots of algebraic equations. Linear and Multilinear Algebra. 1981;10(4):265-308. DOI: 10.1080/03081088108817420\n'},{id:"B14",body:'Chinen K. An abundance of invariant polynomials satisfying the Riemann hypothesis. Discrete Mathematics. 2008;308(24):6426-6440. DOI: 10.1016/j.disc.2007.12.022\n'},{id:"B15",body:'Choo Y. On the zeros of a family of self-reciprocal polynomials. International Journal of Mathematical Analysis. 2011;5(36):1761-1766\n'},{id:"B16",body:'Lakatos P. On zeros of reciprocal polynomials. Publicationes Mathematicae Debrecen. 2002;61(3–4):645-661\n'},{id:"B17",body:'Lakatos P, Losonczi L. On zeros of reciprocal polynomials of odd degree. Journal of Inequalities in Pure and Applied Mathematics. 2003;4(3):8-15\n'},{id:"B18",body:'Lakatos P, Losonczi L. Circular interlacing with reciprocal polynomials. Mathematical Inequalities & Applications. 2007;10(4):761. DOI: 10.7153/mia-10-71\n'},{id:"B19",body:'Kwon DY. Reciprocal polynomials with all zeros on the unit circle. Acta Mathematica Hungarica. 2011;131(3):285-294. DOI: 10.1007/s10474–011–0090–6\n'},{id:"B20",body:'Kwon DY. Reciprocal polynomials with all but two zeros on the unit circle. Acta Mathematica Hungarica. 2011;134(4):472-480. DOI: 10.1007/s10474–011–0176–1\n'},{id:"B21",body:'Konvalina J, Matache V. Palindrome-polynomials with roots on the unit circle. Comptes Rendus Mathematique. 2004;26(2):39\n'},{id:"B22",body:'Kim S-H, Park CW. On the zeros of certain self-reciprocal polynomials. Journal of Mathematical Analysis and Applications. 2008;339(1):240-247. DOI: 10.1016/j.jmaa.2007.06.055\n'},{id:"B23",body:'Kim S-H, Lee JH. On the zeros of self-reciprocal polynomials satisfying certain coefficient conditions. Bulletin of the Korean Mathematical Society. 2010;47(6):1189-1194. DOI: 10.4134/BKMS.2010.47.6.1189\n'},{id:"B24",body:'Botta V, Bracciali CF, Pereira JA. Some properties of classes of real self-reciprocal polynomials. Journal of Mathematical Analysis and Applications. 2016;433(2):1290-1304. DOI: 10.1016/j.jmaa.2015.08.038\n'},{id:"B25",body:'Suzuki M. On zeros of self-reciprocal polynomials. 2012. ArXiv: ArXiv:1211.2953\n'},{id:"B26",body:'Botta V, Marques LF, Meneguette M. Palindromic and perturbed polynomials: Zeros location. Acta Mathematica Hungarica. 2014;143(1):81-87. DOI: 10.1007/s10474–013–0382–0\n'},{id:"B27",body:'Conrey B, Granville A, Poonen B, Soundararajan K. Zeros of Fekete polynomials. Annales de l’institut Fourier. 2000;50(3):865-890. DOI: 10.5802/aif.1776\n'},{id:"B28",body:'Erdélyi T. On the zeros of polynomials with Littlewood-type coefficient constraints. The Michigan Mathematical Journal. 2001;49(1):97-111. DOI: 10.1307/mmj/1008719037\n'},{id:"B29",body:'Mossinghoff MJ. Polynomials with restricted coefficients and prescribed non-cyclotomic factors. LMS Journal of Computation and Mathematics. 2003;6:314-325. DOI: 10.1112/S1461157000000474\n'},{id:"B30",body:'Mercer ID. Unimodular roots of special Littlewood polynomials. Canadian Mathematical Bulletin. 2006;49(3):438-447. DOI: 10.4153/CMB-2006–043-x\n'},{id:"B31",body:'Mukunda K. Littlewood Pisot numbers. Journal of Number Theory. 2006;117(1):106-121. DOI: 10.1016/j.jnt.2005.05.009\n'},{id:"B32",body:'Drungilas P. Unimodular roots of reciprocal Littlewood polynomials. Journal of the Korean Mathematical Society. 2008;45(3):835-840. DOI: 10.4134/JKMS.2008.45.3.835\n'},{id:"B33",body:'Baradaran J, Taghavi M. Polynomials with coefficients from a finite set. Mathematica Slovaca. 2014;64(6):1397-1408. DOI: 10.2478/s12175–014–0282-y\n'},{id:"B34",body:'Borwein P, Choi S, Ferguson R, Jankauskas J. On Littlewood polynomials with prescribed number of zeros inside the unit disk. Canadian Journal of Mathematics. 2015;67(3):507-526. DOI: 10.4153/CJM-2014–007–1\n'},{id:"B35",body:'Drungilas P, Jankauskas J, Šiurys J. On Littlewood and Newman polynomial multiples of Borwein polynomials. Mathematics of Computation. 2018;87(311):1523-1541. DOI: 10.1090/mcom/3258\n'},{id:"B36",body:'Odlyzko A, Poonen B. Zeros of polynomials with 0,1 coefficients. L’Enseignement Mathématique. 1993;39:317-348\n'},{id:"B37",body:'Murty MR, Smyth C, Wang RJ. Zeros of Ramanujan polynomials. Journal of the Ramanujan Mathematical Society. 2011;26(1):107-125\n'},{id:"B38",body:'Lalín MN, Rogers MD, et al. Variations of the Ramanujan polynomials and remarks on \n\nζ\n\n\n2\nj\n+\n1\n\n\n/\n\nπ\n\n2\nj\n+\n1\n\n\n\n. Functiones et Approximatio Commentarii Mathematici. 2013;48(1):91-111. DOI: 10.7169/facm/2013.48.1.8\n'},{id:"B39",body:'Diamantis N, Rolen L. Period polynomials, derivatives of L-functions, and zeros of polynomials. Research in the Mathematical Sciences. 2018;5(1):9. DOI: 10.1007/s40687–018–0126–4\n'},{id:"B40",body:'Lindstrøm P. Galois Theory of Palindromic Polynomials. Oslo: University of Oslo; 2015\n'},{id:"B41",body:'O’Hara PJ, Rodriguez RS. Some properties of self-inversive polynomials. Proceedings of the American Mathematical Society. 1974;44(2):331-335. DOI: 10.1090/S0002–9939–1974–0349967–5\n'},{id:"B42",body:'Schinzel A. Self-inversive polynomials with all zeros on the unit circle. The Ramanujan Journal. 2005;9(1):19-23. DOI: 10.1007/s11139–005–0821–9\n'},{id:"B43",body:'Losonczi L, Schinzel A. Self-inversive polynomials of odd degree. The Ramanujan Journal. 2007;14(2):305-320. DOI: 10.1007/s11139–007–9029–5\n'},{id:"B44",body:'Lakatos P, Losonczi L. Self-inversive polynomials whose zeros are on the unit circle. Publicationes Mathematicae Debrecen. 2004;65(3–4):409-420\n'},{id:"B45",body:'Lakatos P, Losonczi L. Polynomials with all zeros on the unit circle. Acta Mathematica Hungarica. 2009;125(4):341-356. DOI: 10.1007/s10474–009–9028–7\n'},{id:"B46",body:'Losonczi L. On reciprocal polynomials with zeros of modulus one. Mathematical Inequalities & Applications. 2006;9(2):289. DOI: 10.7153/mia-09–29\n'},{id:"B47",body:'Vieira RS. On the number of roots of self-inversive polynomials on the complex unit circle. The Ramanujan Journal. 2017;42(2):363-369. DOI: 10.1007/s11139–016–9804–2\n'},{id:"B48",body:'Stankov D. The necessary and sufficient condition for an algebraic integer to be a Salem number. 2018. arXiv:1706.01767\n'},{id:"B49",body:'Sinclair C, Vaaler J. Self-inversive polynomials with all zeros on the unit circle. In: McKee J, Smyth C, editors. Number Theory and Polynomials. London Mathematical Society Lecture Note Series. Cambridge: Cambridge University Press; 2008. pp. 312-321. DOI: 10.1017/CB09780511721274.020\n'},{id:"B50",body:'Choo Y, Kim Y-J. On the zeros of self-inversive polynomials. International Journal of Mathematical Analysis. 2013;7:187-193\n'},{id:"B51",body:'Area I, Godoy E, Lamblém RL, Ranga AS. Basic hypergeometric polynomials with zeros on the unit circle. Applied Mathematics and Computation. 2013;225:622-630. DOI: 10.1016/j.amc.2013.09.060\n'},{id:"B52",body:'Dimitrov D, Ismail M, Ranga AS. A class of hypergeometric polynomials with zeros on the unit circle: Extremal and orthogonal properties and quadrature formulas. Applied Numerical Mathematics. 2013;65:41-52\n'},{id:"B53",body:'Kim E. A family of self-inversive polynomials with concyclic zeros. Journal of Mathematical Analysis and Applications. 2013;401(2):695-701. DOI: 10.1016/j.jmaa.2012.12.048\n'},{id:"B54",body:'Ito N, Wimmer HK. Self-inversive Hilbert space operator polynomials with spectrum on the unit circle. Journal of Mathematical Analysis and Applications. 2016;436(2):683-691. DOI: 10.4153/CMB-2001–044-x\n'},{id:"B55",body:'Joyner D, Kim J-L. Selected Unsolved Problems in Coding Theory. Birkhäuser Basel, New York: Springer Science & Business Media; 2011. DOI: 10.1007/978–0-8176–8256–9\n'},{id:"B56",body:'Joyner D. Zeros of some self-reciprocal polynomials. In: Excursions in Harmonic Analysis. Vol. 1. Birkhäuser, Boston: Springer; 2013. pp. 329-348. DOI: 10.1007/978–0-8176–8376–4_17\n'},{id:"B57",body:'Joyner D, Shaska T. Self-inversive polynomials, curves, and codes. In: Higher Genus Curves in Mathematical Physics and Arithmetic Geometry. Vol. 703. American Mathematical Society; 2018. pp. 189-208. DOI: 10.1090/conm/703\n'},{id:"B58",body:'McKee J, McKee JF, Smyth C. Number Theory and Polynomials. Vol. 352. Cambridge: Cambridge University Press; 2008\n'},{id:"B59",body:'Lee T-D, Yang C-N. Statistical theory of equations of state and phase transitions II. Lattice gas and Ising model. Physical Review. 1952;87(3):410. DOI: 10.1103/PhysRev.87.410\n'},{id:"B60",body:'He Y-H. Polynomial roots and Calabi-Yau geometries. Advances in High Energy Physics. 2011;2011:1-15. DOI: 10.1155/2011/719672\n'},{id:"B61",body:'Everest G, Ward T. Heights of Polynomials and Entropy in Algebraic Dynamics. Springer-Verlag, London: Springer Science & Business Media; 2013\n'},{id:"B62",body:'Bertin MJ, Decomps-Guilloux A, Grandet-Hugot M, Pathiaux-Delefosse M, Schreiber J. Pisot and Salem Numbers. Birkhäuser, Basel: Birkhäuser; 2012\n'},{id:"B63",body:'Smyth CJ. On the product of the conjugates outside the unit circle of an algebraic integer. Bulletin of the London Mathematical Society. 1971;3(2):169-175. DOI: 10.1112/blms/3.2.169\n'},{id:"B64",body:'Smyth C. Seventy years of Salem numbers. Bulletin of the London Mathematical Society. 2015;47(3):379-395. DOI: 10.1112/blms/bdv027\n'},{id:"B65",body:'Boyd DW. Small Salem numbers. Duke Mathematical Journal. 1977;44(2):315-328. DOI: 10.1215/S0012–7094–77–04413–1\n'},{id:"B66",body:'Mossinghoff M. Polynomials with small Mahler measure. Mathematics of Computation of the American Mathematical Society. 1998;67(224):1697-1705. DOI: 10.1090/S0025–5718–98–01006–0\n'},{id:"B67",body:'Lehmer DH. Factorization of certain cyclotomic functions. Annals of Mathematics. 1933;34(3):461-479. DOI: 10.2307/1968172\n'},{id:"B68",body:'Hironaka E. What is… Lehmer’s number? Notices of the American Mathematical Society; 2009;56:374-375\n'},{id:"B69",body:'Bailey DH, Broadhurst DJ. A seventeenth-order polylogarithm ladder. 1999. arXiv preprint: math/9906134\n'},{id:"B70",body:'Adams CC. The Knot Book: An Elementary Introduction to the Mathematical Theory of Knots. Providence: American Mathematical Society; 2004\n'},{id:"B71",body:'Alexander JW. Topological invariants of knots and links. Transactions of the American Mathematical Society. 1928;30(2):275-306. DOI: 10.1090/S0002–9947–1928–1501429–1\n'},{id:"B72",body:'Jones V. A polynomial invariant for knots via von Neumann algebras. Bulletin of the American Mathematical Society. 1985;12:103-111\n'},{id:"B73",body:'Hironaka E. The Lehmer polynomial and pretzel links. Canadian Mathematical Bulletin. 2001;44(4):440-451\n'},{id:"B74",body:'Vieira R. Solving and classifying the solutions of the Yang-Baxter equation through a differential approach. Two-state systems. Journal of High Energy Physics. 2018;2018(10):110. DOI: 10.1007/JHEP10(2018)110\n'},{id:"B75",body:'Bethe H. Zur theorie der metalle. Zeitschrift für Physik. 1931;71(3–4):205-226\n'},{id:"B76",body:'Vieira RS, Lima-Santos A. Where are the roots of the Bethe Ansatz equations? Physics Letters A. 2015;379(37):2150-2153. DOI: 10.1016/j.physleta.2015.07.016\n'},{id:"B77",body:'Chihara TS. An Introduction to Orthogonal Polynomials. Dover Publications; 2011\n'},{id:"B78",body:'Vieira RS, Botta V. Möbius transformations and orthogonal polynomials (In preparation). New York: Gordon and Breach Science Publishers; 1978\n'},{id:"B79",body:'Vieira RS, Botta V. Möbius transformations, orthogonal polynomials and self-inversive polynomials (In preparation). New York: Gordon and Breach Science Publishers; 1978\n'}],footnotes:[{id:"fn1",explanation:"The reader should be aware that there is no standard in naming these polynomials. For instance, what we call here self-inversive polynomials are sometimes called self-reciprocal polynomials. What we mean positive self-reciprocal polynomials are usually just called self-reciprocal or yet palindrome polynomials (because their coefficients are the same whether they are read from forwards or backwards), as well as, negative self-reciprocal polynomials are usually called skew-reciprocal, anti-reciprocal, or yet anti-palindrome polynomials."},{id:"fn2",explanation:"The zeros of such polynomials present a fractal behavior, as was first discovered by Odlyzko and Poonen in [36]."},{id:"fn3",explanation:"The Mahler measure of a monic integer polynomial \n\np\n\nz\n\n\n can also be defined without making reference to its zeros through the formula \n\nM\n\n\np\n\nz\n\n\n\n=\nexp\n\n\n\n∫\n0\n1\n\nlog\n\n\np\n\n\ne\n\n2\nπit\n\n\n\n\n\ndt\n\n\n\n—see [61]."},{id:"fn4",explanation:"We remark, however, that different knots can have the same knot invariant. Up to date, we do not know whether there exists a knot invariant that distinguishes all non-equivalent knots from each other (although there do exist some invariants that distinguish every knot from the trivial knot). Thus, until now the concept of knot invariants only partially solves the problem of knot classification."},{id:"fn5",explanation:"Alexander polynomials can also be defined as Laurent polynomials, see [70]."}],contributors:[{corresp:"yes",contributorFullName:"Ricardo Vieira",address:"rs.vieira@unesp.br",affiliation:'
Faculty of Science and Technology, Department of Mathematics and Computer Science, São Paulo State University (UNESP), Presidente Prudente, SP, Brazil
'}],corrections:null},book:{id:"8599",title:"Polynomials",subtitle:"Theory and Application",fullTitle:"Polynomials - Theory and Application",slug:"polynomials-theory-and-application",publishedDate:"May 2nd 2019",bookSignature:"Cheon Seoung Ryoo",coverURL:"https://cdn.intechopen.com/books/images_new/8599.jpg",licenceType:"CC BY 3.0",editedByType:"Edited by",editors:[{id:"230100",title:"Prof.",name:"Cheon Seoung",middleName:null,surname:"Ryoo",slug:"cheon-seoung-ryoo",fullName:"Cheon Seoung Ryoo"}],productType:{id:"1",title:"Edited Volume",chapterContentType:"chapter",authoredCaption:"Edited by"},chapters:[{id:"66597",title:"Cyclotomic and Littlewood Polynomials Associated to Algebras",slug:"cyclotomic-and-littlewood-polynomials-associated-to-algebras",totalDownloads:197,totalCrossrefCites:0,signatures:"José-Antonio de la Peña",authors:[null]},{id:"64494",title:"New Aspects of Descartes’ Rule of Signs",slug:"new-aspects-of-descartes-rule-of-signs",totalDownloads:259,totalCrossrefCites:0,signatures:"Vladimir Petrov Kostov and Boris Shapiro",authors:[null]},{id:"64679",title:"Obtaining Explicit Formulas and Identities for Polynomials Defined by Generating Functions of the Form F(t)x ⋅ G(t)α",slug:"obtaining-explicit-formulas-and-identities-for-polynomials-defined-by-generating-functions-of-the-fo",totalDownloads:183,totalCrossrefCites:0,signatures:"Dmitry Kruchinin, Vladimir Kruchinin and Yuriy Shablya",authors:[null]},{id:"66332",title:"Polynomials with Symmetric Zeros",slug:"polynomials-with-symmetric-zeros",totalDownloads:250,totalCrossrefCites:0,signatures:"Ricardo Vieira",authors:[null]},{id:"65322",title:"A Numerical Investigation on the Structure of the Zeros of the Q-Tangent Polynomials",slug:"a-numerical-investigation-on-the-structure-of-the-zeros-of-the-q-tangent-polynomials",totalDownloads:207,totalCrossrefCites:0,signatures:"Jung Yoog Kang and Cheon Seoung Ryoo",authors:[null]},{id:"65970",title:"Investigation and Synthesis of Robust Polynomials in Uncertainty on the Basis of the Root Locus Theory",slug:"investigation-and-synthesis-of-robust-polynomials-in-uncertainty-on-the-basis-of-the-root-locus-theo",totalDownloads:214,totalCrossrefCites:0,signatures:"Nesenchuk Alla",authors:[null]},{id:"65230",title:"Pricing Basket Options by Polynomial Approximations",slug:"pricing-basket-options-by-polynomial-approximations",totalDownloads:189,totalCrossrefCites:0,signatures:"Pablo Olivares",authors:[null]},{id:"65150",title:"The Orthogonal Expansion in Time-Domain Method for Solving Maxwell Equations Using Paralleling-in-Order Scheme",slug:"the-orthogonal-expansion-in-time-domain-method-for-solving-maxwell-equations-using-paralleling-in-or",totalDownloads:204,totalCrossrefCites:0,signatures:"Zheng-Yu Huang, Zheng Sun and Wei He",authors:[null]}]},relatedBooks:[{type:"book",id:"3161",title:"Frontiers in Guided Wave Optics and Optoelectronics",subtitle:null,isOpenForSubmission:!1,hash:"deb44e9c99f82bbce1083abea743146c",slug:"frontiers-in-guided-wave-optics-and-optoelectronics",bookSignature:"Bishnu Pal",coverURL:"https://cdn.intechopen.com/books/images_new/3161.jpg",editedByType:"Edited by",editors:[{id:"4782",title:"Prof.",name:"Bishnu",surname:"Pal",slug:"bishnu-pal",fullName:"Bishnu Pal"}],productType:{id:"1",chapterContentType:"chapter",authoredCaption:"Edited by"},chapters:[{id:"8425",title:"Frontiers in Guided Wave Optics and Optoelectronics",slug:"frontiers-in-guided-wave-optics-and-optoelectronics",signatures:"Bishnu Pal",authors:[{id:"4782",title:"Prof.",name:"Bishnu",middleName:"P",surname:"Pal",fullName:"Bishnu Pal",slug:"bishnu-pal"}]},{id:"8426",title:"Application Specific Optical Fibers",slug:"application-specific-optical-fibers",signatures:"Bishnu P. Pal",authors:[{id:"4782",title:"Prof.",name:"Bishnu",middleName:"P",surname:"Pal",fullName:"Bishnu Pal",slug:"bishnu-pal"}]},{id:"8427",title:"Nonlinear Properties of Chalcogenide Glass Fibers",slug:"nonlinear-properties-of-chalcogenide-glass-fibers",signatures:"Jas S. Sanghera, L. Brandon Shaw, C. M. Florea, P. Pureza, V. Q. Nguyen, F. Kung, Dan Gibson and I. D. Aggarwal",authors:[{id:"5111",title:"Dr.",name:"Jasbinder",middleName:null,surname:"Sanghera",fullName:"Jasbinder Sanghera",slug:"jasbinder-sanghera"},{id:"133867",title:"Dr.",name:"Brandon",middleName:null,surname:"Shaw",fullName:"Brandon Shaw",slug:"brandon-shaw"},{id:"133868",title:"Dr.",name:"Catalin",middleName:null,surname:"Florea",fullName:"Catalin Florea",slug:"catalin-florea"},{id:"133872",title:"Prof.",name:"Gam",middleName:null,surname:"Nguyen",fullName:"Gam Nguyen",slug:"gam-nguyen"},{id:"133876",title:"Dr.",name:"Ishwar",middleName:null,surname:"Aggarwal",fullName:"Ishwar Aggarwal",slug:"ishwar-aggarwal"}]},{id:"8428",title:"Irradiation Effects in Optical Fibers",slug:"irradiation-effects-in-optical-fibers",signatures:"Sporea Dan, Agnello Simonpietro and Gelardi Franco Mario",authors:[{id:"5392",title:"Dr.",name:"Dan",middleName:null,surname:"Sporea",fullName:"Dan Sporea",slug:"dan-sporea"},{id:"133835",title:"Prof.",name:"Simonpietro",middleName:null,surname:"Agnello",fullName:"Simonpietro Agnello",slug:"simonpietro-agnello"},{id:"133836",title:"Prof.",name:"Franco Mario",middleName:null,surname:"Gelardi",fullName:"Franco Mario Gelardi",slug:"franco-mario-gelardi"}]},{id:"8429",title:"Programmable All-Fiber Optical Pulse Shaping",slug:"programmable-all-fiber-optical-pulse-shaping",signatures:"Antonio Malacarne, Saju Thomas, Francesco Fresi, Luca Potì, Antonella Bogoni and Josè Azaña",authors:[{id:"2707",title:"Dr.",name:"Luca",middleName:null,surname:"Poti",fullName:"Luca Poti",slug:"luca-poti"},{id:"4930",title:"Dr.",name:"Antonio",middleName:null,surname:"Malacarne",fullName:"Antonio Malacarne",slug:"antonio-malacarne"},{id:"5391",title:"Dr.",name:"Francesco",middleName:null,surname:"Fresi",fullName:"Francesco Fresi",slug:"francesco-fresi"},{id:"109285",title:"Dr.",name:"Antonella",middleName:null,surname:"Bogoni",fullName:"Antonella Bogoni",slug:"antonella-bogoni"},{id:"133856",title:"Prof.",name:"Thomas",middleName:null,surname:"Saju",fullName:"Thomas Saju",slug:"thomas-saju"},{id:"133858",title:"Prof.",name:"Jose",middleName:null,surname:"Azana",fullName:"Jose Azana",slug:"jose-azana"}]},{id:"8430",title:"Physical Nature of “Slow Light” in Stimulated Brillouin Scattering",slug:"physical-nature-of-slow-light-in-stimulated-brillouin-scattering",signatures:"Valeri I. Kovalev, Robert G. Harrison and Nadezhda E. Kotova",authors:[{id:"4873",title:"Dr.",name:"Valeri",middleName:null,surname:"Kovalev",fullName:"Valeri Kovalev",slug:"valeri-kovalev"},{id:"133838",title:"Prof.",name:"Robert",middleName:null,surname:"Harrison",fullName:"Robert Harrison",slug:"robert-harrison"}]},{id:"8431",title:"Bismuth-doped Silica Fiber Amplifier",slug:"bismuth-doped-silica-fiber-amplifier",signatures:"Young-Seok Seo and Yasushi Fujimoto",authors:[{id:"4778",title:"Researcher",name:"Young-Seok",middleName:null,surname:"Seo",fullName:"Young-Seok Seo",slug:"young-seok-seo"},{id:"4885",title:"Dr.",name:"Yasushi",middleName:null,surname:"Fujimoto",fullName:"Yasushi Fujimoto",slug:"yasushi-fujimoto"}]},{id:"8432",title:"Radio-over-Fibre Techniques and Performance",slug:"radio-over-fibre-techniques-and-performance",signatures:"Roberto Llorente and Marta Beltrán",authors:[{id:"4404",title:"Ms.",name:"Marta",middleName:null,surname:"Beltran",fullName:"Marta Beltran",slug:"marta-beltran"},{id:"16540",title:"Dr.",name:"Roberto",middleName:null,surname:"Llorente",fullName:"Roberto Llorente",slug:"roberto-llorente"}]},{id:"8433",title:"Time-Spectral Visualization of Fundamental Ultrafast Nonlinear-Optical Interactions in Photonic Fibers",slug:"time-spectral-visualization-of-fundamental-ultrafast-nonlinear-optical-interactions-in-photonic-fibe",signatures:"Anatoly Efimov",authors:[{id:"4545",title:"Dr.",name:"Anatoly",middleName:null,surname:"Efimov",fullName:"Anatoly Efimov",slug:"anatoly-efimov"}]},{id:"8434",title:"Dispersion Compensation Devices",slug:"dispersion-compensation-devices",signatures:"Lingling Chen, Meng Zhang and Zhigang Zhang",authors:[{id:"4565",title:"Ms.",name:"Lingling",middleName:null,surname:"Chen",fullName:"Lingling Chen",slug:"lingling-chen"},{id:"4773",title:"Professor",name:"Zhigang",middleName:null,surname:"Zhang",fullName:"Zhigang Zhang",slug:"zhigang-zhang"}]},{id:"8435",title:"Photonic Crystal Fibre for Dispersion Controll",slug:"photonic-crystal-fibre-for-dispersion-controll",signatures:"Zoltán Várallyay and Kunimasa Saitoh",authors:[{id:"4607",title:"Dr.",name:"Zoltan Krisztian",middleName:null,surname:"Varallyay",fullName:"Zoltan Krisztian Varallyay",slug:"zoltan-krisztian-varallyay"},{id:"133834",title:"Prof.",name:"Kunimasa",middleName:null,surname:"Saitoh",fullName:"Kunimasa Saitoh",slug:"kunimasa-saitoh"}]},{id:"8436",title:"Resonantly Induced Refractive Index Changes in Yb-doped Fibers: the Origin, Properties and Application for All-Fiber Coherent Beam Combining",slug:"resonantly-induced-refractive-index-changes-in-yb-doped-fibers-the-origin-properties-and-application",signatures:"Andrei A. Fotiadi, Oleg L. Antipov and Patrice Mégret",authors:[{id:"4725",title:"Dr.",name:"Andrei",middleName:null,surname:"Fotiadi",fullName:"Andrei Fotiadi",slug:"andrei-fotiadi"},{id:"107849",title:"Prof.",name:"Patrice",middleName:null,surname:"Mégret",fullName:"Patrice Mégret",slug:"patrice-megret"},{id:"133847",title:"Prof.",name:"Oleg",middleName:null,surname:"Antipov",fullName:"Oleg Antipov",slug:"oleg-antipov"}]},{id:"8437",title:"Polarization Coupling of Light and Optoelectronics Devices Based on Periodically Poled Lithium Niobate",slug:"polarization-coupling-of-light-and-optoelectronics-devices-based-on-periodically-poled-lithium-nioba",signatures:"Xianfeng Chen, Kun Liu, and Jianhong Shi",authors:[{id:"4180",title:"Professor",name:"Xianfeng",middleName:null,surname:"Chen",fullName:"Xianfeng Chen",slug:"xianfeng-chen"},{id:"133851",title:"Prof.",name:"Kun",middleName:null,surname:"Liu",fullName:"Kun Liu",slug:"kun-liu"},{id:"133853",title:"Prof.",name:"Jianhong",middleName:null,surname:"Shi",fullName:"Jianhong Shi",slug:"jianhong-shi"}]},{id:"8438",title:"All-Optical Wavelength-Selective Switch by Intensity Control in Cascaded Interferometers",slug:"all-optical-wavelength-selective-switch-by-intensity-control-in-cascaded-interferometers",signatures:"Hiroki Kishikawa, Nobuo Goto and Kenta Kimiya",authors:[{id:"4400",title:"Professor",name:"Nobuo",middleName:null,surname:"Goto",fullName:"Nobuo Goto",slug:"nobuo-goto"},{id:"133356",title:"Prof.",name:"Hiroki",middleName:null,surname:"Kishikawa",fullName:"Hiroki Kishikawa",slug:"hiroki-kishikawa"},{id:"133358",title:"Prof.",name:"Kenta",middleName:null,surname:"Kimiya",fullName:"Kenta Kimiya",slug:"kenta-kimiya"}]},{id:"8439",title:"Nonlinear Optics in Doped Silica Glass Integrated Waveguide Structures",slug:"nonlinear-optics-in-doped-silica-glass-integrated-waveguide-structures",signatures:"David Duchesne, Marcello Ferrera, Luca Razzari, Roberto Morandotti, Brent Little, Sai T. Chu and David J. Moss",authors:[{id:"4405",title:"Dr.",name:"David",middleName:null,surname:"Moss",fullName:"David Moss",slug:"david-moss"},{id:"4783",title:"Dr.",name:"David",middleName:null,surname:"Duchesne",fullName:"David Duchesne",slug:"david-duchesne"},{id:"95840",title:"Dr.",name:"Luca",middleName:null,surname:"Razzari",fullName:"Luca Razzari",slug:"luca-razzari"},{id:"135390",title:"Prof.",name:"Marcello",middleName:null,surname:"Ferrera",fullName:"Marcello Ferrera",slug:"marcello-ferrera"},{id:"135391",title:"Prof.",name:"Roberto",middleName:null,surname:"Morandotti",fullName:"Roberto Morandotti",slug:"roberto-morandotti"},{id:"135392",title:"Prof.",name:"Brent",middleName:null,surname:"Little",fullName:"Brent Little",slug:"brent-little"},{id:"135393",title:"Prof.",name:"Sai",middleName:null,surname:"Chu",fullName:"Sai Chu",slug:"sai-chu"}]},{id:"8440",title:"Advances in Femtosecond Micromachining and Inscription of Micro and Nano Photonic Devices",slug:"advances-in-femtosecond-micromachining-and-inscription-of-micro-and-nano-photonic-devices",signatures:"Graham N. Smith, Kyriacos Kalli and Kate Sugden",authors:[{id:"4668",title:"Dr.",name:"Graham",middleName:"N",surname:"Smith",fullName:"Graham Smith",slug:"graham-smith"},{id:"133360",title:"Prof.",name:"Kyriacos",middleName:null,surname:"Kalli",fullName:"Kyriacos Kalli",slug:"kyriacos-kalli"},{id:"133361",title:"Prof.",name:"Kate",middleName:null,surname:"Sugden",fullName:"Kate Sugden",slug:"kate-sugden"}]},{id:"8441",title:"Magneto-Optical Devices for Optical Integrated Circuits",slug:"magneto-optical-devices-for-optical-integrated-circuits",signatures:"Vadym Zayets and Koji Ando",authors:[{id:"4688",title:"Dr.",name:"Vadym",middleName:null,surname:"Zayets",fullName:"Vadym Zayets",slug:"vadym-zayets"},{id:"133363",title:"Prof.",name:"Koji",middleName:null,surname:"Ando",fullName:"Koji Ando",slug:"koji-ando"}]},{id:"8442",title:"Tunable Hollow Optical Waveguide and Its Applications",slug:"tunable-hollow-optical-waveguide-and-its-applications",signatures:"Mukesh Kumar, Toru Miura, Yasuki Sakurai and Fumio Koyama",authors:[{id:"63461",title:"Dr.",name:"Mukesh",middleName:null,surname:"Kumar",fullName:"Mukesh Kumar",slug:"mukesh-kumar"},{id:"133388",title:"Prof.",name:"Toru",middleName:null,surname:"Miura",fullName:"Toru Miura",slug:"toru-miura"},{id:"133402",title:"Prof.",name:"Yasuki",middleName:null,surname:"Sakurai",fullName:"Yasuki Sakurai",slug:"yasuki-sakurai"},{id:"133404",title:"Prof.",name:"Fumio",middleName:null,surname:"Koyama",fullName:"Fumio Koyama",slug:"fumio-koyama"}]},{id:"8443",title:"Regenerated Fibre Bragg Gratings",slug:"regenerated-fibre-bragg-gratings",signatures:"John Canning, Somnath Bandyopadhyay, Palas Biswas, Mattias Aslund, Michael Stevenson and Kevin Cook",authors:[{id:"5461",title:"Professor",name:"John",middleName:null,surname:"Canning",fullName:"John Canning",slug:"john-canning"},{id:"133394",title:"Dr.",name:"Somnath",middleName:null,surname:"Bandyopadhyay",fullName:"Somnath Bandyopadhyay",slug:"somnath-bandyopadhyay"},{id:"133395",title:"Prof.",name:"Palas",middleName:null,surname:"Biswas",fullName:"Palas Biswas",slug:"palas-biswas"},{id:"133396",title:"Prof.",name:"Mattias",middleName:null,surname:"Aslund",fullName:"Mattias Aslund",slug:"mattias-aslund"},{id:"133397",title:"Prof.",name:"Michael",middleName:null,surname:"Stevenson",fullName:"Michael Stevenson",slug:"michael-stevenson"},{id:"133400",title:"Prof.",name:"Kevin",middleName:null,surname:"Cook",fullName:"Kevin Cook",slug:"kevin-cook"}]},{id:"8444",title:"Optical Deposition of Carbon Nanotubes for Fiber-based Device Fabrication",slug:"optical-deposition-of-carbon-nanotubes-for-fiber-based-device-fabrication",signatures:"Ken Kashiwagi and Shinji Yamashita",authors:[{id:"5133",title:"Dr.",name:"Ken",middleName:null,surname:"Kashiwagi",fullName:"Ken Kashiwagi",slug:"ken-kashiwagi"},{id:"38416",title:"Mr.",name:"Shinji",middleName:null,surname:"Yamashita",fullName:"Shinji Yamashita",slug:"shinji-yamashita"}]},{id:"8445",title:"High Power Tunable Tm3+-fiber Lasers and Its Application in Pumping Cr2+:ZnSe Lasers",slug:"high-power-tunable-tm3-fiber-lasers-and-its-application-in-pumping-cr2-znse-lasers",signatures:"Yulong Tang and Jianqiu Xu",authors:[{id:"5449",title:"Prof.",name:"Jianqiu",middleName:null,surname:"Xu",fullName:"Jianqiu Xu",slug:"jianqiu-xu"},{id:"110808",title:"Dr.",name:"Yulong",middleName:null,surname:"Tang",fullName:"Yulong Tang",slug:"yulong-tang"}]},{id:"8446",title:"2 µm Laser Sources and Their Possible Applications",slug:"2-m-laser-sources-and-their-possible-applications",signatures:"Karsten Scholle, Samir Lamrini, Philipp Koopmann and Peter Fuhrberg",authors:[{id:"4951",title:"Dr.",name:"Karsten",middleName:null,surname:"Scholle",fullName:"Karsten Scholle",slug:"karsten-scholle"},{id:"133366",title:"Prof.",name:"Samir",middleName:null,surname:"Lamrini",fullName:"Samir Lamrini",slug:"samir-lamrini"},{id:"133370",title:"Prof.",name:"Philipp",middleName:null,surname:"Koopmann",fullName:"Philipp Koopmann",slug:"philipp-koopmann"},{id:"133371",title:"Mr.",name:"Peter",middleName:null,surname:"Fuhrberg",fullName:"Peter Fuhrberg",slug:"peter-fuhrberg"}]},{id:"8447",title:"Designer Laser Resonators based on Amplifying Photonic Crystals",slug:"designer-laser-resonators-based-on-amplifying-photonic-crystals",signatures:"Alexander Benz, Christoph Deutsch, Gernot Fasching, Karl Unterrainer, Aaron M. Maxwell, Pavel Klang, Werner Schrenk and Gottfried Strasser",authors:[{id:"4537",title:"DI",name:"Alexander",middleName:null,surname:"Benz",fullName:"Alexander Benz",slug:"alexander-benz"},{id:"135394",title:"Prof.",name:"Christoph",middleName:null,surname:"Deutsch",fullName:"Christoph Deutsch",slug:"christoph-deutsch"},{id:"135395",title:"Prof.",name:"Gernot",middleName:null,surname:"Fasching",fullName:"Gernot Fasching",slug:"gernot-fasching"},{id:"135396",title:"Prof.",name:"Karl",middleName:null,surname:"Unterrainer",fullName:"Karl Unterrainer",slug:"karl-unterrainer"},{id:"135397",title:"Prof.",name:"Aaron",middleName:null,surname:"Maxwell",fullName:"Aaron Maxwell",slug:"aaron-maxwell"},{id:"135398",title:"Prof.",name:"Pavel",middleName:null,surname:"Klang",fullName:"Pavel Klang",slug:"pavel-klang"},{id:"135399",title:"Prof.",name:"Werner",middleName:null,surname:"Schrenk",fullName:"Werner Schrenk",slug:"werner-schrenk"},{id:"135400",title:"Prof.",name:"Gottfried",middleName:null,surname:"Strasser",fullName:"Gottfried Strasser",slug:"gottfried-strasser"}]},{id:"8448",title:"High-Power and High Efficiency Yb:YAG Ceramic Laser at Room Temperature",slug:"high-power-and-high-efficiency-yb-yag-ceramic-laser-at-room-temperature",signatures:"Shinki Nakamura",authors:[{id:"4143",title:"Dr.",name:"Shinki",middleName:null,surname:"Nakamura",fullName:"Shinki Nakamura",slug:"shinki-nakamura"}]},{id:"8449",title:"Polarization Properties of Laser-Diode-Pumped Microchip Nd:YAG Ceramic Lasers",slug:"polarization-properties-of-laser-diode-pumped-microchip-nd-yag-ceramic-lasers",signatures:"Kenju Otsuka",authors:[{id:"4259",title:"Professor",name:"Kenju",middleName:null,surname:"Otsuka",fullName:"Kenju Otsuka",slug:"kenju-otsuka"}]},{id:"8450",title:"Surface-Emitting Circular Bragg Lasers – A Promising Next-Generation On-Chip Light Source for Optical Communications",slug:"surface-emitting-circular-bragg-lasers-a-promising-next-generation-on-chip-light-source-for-optical-",signatures:"Xiankai Sun and Amnon Yariv",authors:[{id:"4201",title:"Prof.",name:"Xiankai",middleName:null,surname:"Sun",fullName:"Xiankai Sun",slug:"xiankai-sun"},{id:"122981",title:"Dr.",name:"Amnon",middleName:null,surname:"Yariv",fullName:"Amnon Yariv",slug:"amnon-yariv"}]},{id:"8451",title:"Novel Enabling Technologies for Convergence of Optical and Wireless Access Networks",slug:"novel-enabling-technologies-for-convergence-of-optical-and-wireless-access-networks",signatures:"Jianjun Yu, Gee-Kung Chang, Zhensheng Jia and Lin Chen",authors:[{id:"8503",title:"Dr.",name:"Jianjun",middleName:null,surname:"Yu",fullName:"Jianjun Yu",slug:"jianjun-yu"},{id:"133376",title:"Prof.",name:"Gee-Kung",middleName:null,surname:"Chang",fullName:"Gee-Kung Chang",slug:"gee-kung-chang"},{id:"133378",title:"Prof.",name:"Zhensheng",middleName:null,surname:"Jia",fullName:"Zhensheng Jia",slug:"zhensheng-jia"},{id:"139599",title:"Prof.",name:"Lin",middleName:null,surname:"Chen",fullName:"Lin Chen",slug:"lin-chen"}]},{id:"8452",title:"Photonic Crystal Multiplexer/Demultiplexer Device for Optical Communications",slug:"photonic-crystal-multiplexer-demultiplexer-device-for-optical-communications",signatures:"Sahbuddin Shaari and Azliza J. M. Adnan",authors:[{id:"19951",title:"Dr.",name:"Sahbudin",middleName:null,surname:"Shaari",fullName:"Sahbudin Shaari",slug:"sahbudin-shaari"}]},{id:"8453",title:"Improvement Scheme for Directly Modulated Fiber Optical CATV System Performances",slug:"improvement-scheme-for-directly-modulated-fiber-optical-catv-system-performances",signatures:"Hai-Han Lu, Ching-Hung Chang and Peng-Chun Peng",authors:[{id:"4684",title:"Professor",name:"Hai-Han",middleName:null,surname:"Lu",fullName:"Hai-Han Lu",slug:"hai-han-lu"},{id:"62688",title:"Prof.",name:"Peng-Chun",middleName:null,surname:"Peng",fullName:"Peng-Chun Peng",slug:"peng-chun-peng"}]},{id:"8454",title:"Optical Beam Steering Using a 2D MEMS Scanner",slug:"optical-beam-steering-using-a-2d-mems-scanner",signatures:"Yves Pétremand, Pierre-André Clerc, Marc Epitaux, Ralf Hauffe, Wilfried Noell and N.F. de Rooij",authors:[{id:"5054",title:"Dr.",name:"Yves",middleName:null,surname:"Petremand",fullName:"Yves Petremand",slug:"yves-petremand"},{id:"135512",title:"Prof.",name:"Pierre-Andre",middleName:null,surname:"Clerc",fullName:"Pierre-Andre Clerc",slug:"pierre-andre-clerc"},{id:"135514",title:"Prof.",name:"Marc",middleName:null,surname:"Epitaux",fullName:"Marc Epitaux",slug:"marc-epitaux"},{id:"135516",title:"Prof.",name:"Ralf",middleName:null,surname:"Hauffe",fullName:"Ralf Hauffe",slug:"ralf-hauffe"},{id:"135518",title:"Prof.",name:"Wilfried",middleName:null,surname:"Noell",fullName:"Wilfried Noell",slug:"wilfried-noell"},{id:"135519",title:"Prof.",name:"N.F.",middleName:null,surname:"De Rooij",fullName:"N.F. De Rooij",slug:"n.f.-de-rooij"}]}]}]},onlineFirst:{chapter:{type:"chapter",id:"66332",title:"Polynomials with Symmetric Zeros",doi:"10.5772/intechopen.82728",slug:"polynomials-with-symmetric-zeros",body:'\n
\n
1. Introduction
\n
In this work, we consider the theory of self-conjugate (SC), self-reciprocal (SR), and self-inversive (SI) polynomials. These are polynomials whose zeros are symmetric either to the real line \n\nR\n\n or to the unit circle \n\nS\n=\n\n\nz\n∈\nC\n:\n\nz\n\n=\n1\n\n\n\n. The basic properties of these polynomials can be found in the books of Marden [1], Milovanović et al. [2], and Sheil-Small [3]. Although these polynomials are very important in both mathematics and physics, it seems that there is no specific review about them; in this work, we present a bird’s eye view to this theory, focusing on the zeros of such polynomials. Other properties of these polynomials (e.g., irreducibility, norms, analytical properties, etc.) are not covered here due to short space, nonetheless, the interested reader can check many of the references presented in the bibliography to this end.
\n
\n
\n
2. Self-conjugate, self-reciprocal, and self-inversive polynomials
\n
We begin with some definitions:
\n
Definition 1. Let \n\np\n\nz\n\n=\n\np\n0\n\n+\n\np\n1\n\nz\n+\n⋯\n+\n\np\n\nn\n−\n1\n\n\n\nz\n\nn\n−\n1\n\n\n+\n\np\nn\n\n\nz\nn\n\n\n be a polynomial of degree \n\nn\n\n with complex coefficients. We shall introduce three polynomials, namely the conjugate polynomial\n\n\np\n¯\n\n\nz\n\n\n, the reciprocal polynomial\n\n\np\n∗\n\n\nz\n\n\n, and the inversive polynomial\n\n\np\n†\n\n\nz\n\n\n, which are, respectively, defined in terms of \n\np\n\nz\n\n\n as follows:
where the bar means complex conjugation. Notice that the conjugate, reciprocal, and inversive polynomials can also be defined without making reference to the coefficients of \n\np\n\nz\n\n\n:
From these relations, we plainly see that if \n\n\nζ\n1\n\n,\n…\n,\n\nζ\nn\n\n\n are the zeros of a complex polynomial \n\np\n\nz\n\n\n of degree \n\nn\n\n, then, the zeros of \n\n\np\n¯\n\n\nz\n\n\n are \n\n\n\nζ\n1\n\n¯\n\n,\n…\n,\n\n\nζ\nn\n\n¯\n\n\n, the zeros of \n\n\np\n∗\n\n\nz\n\n\n are \n\n1\n/\n\nζ\n1\n\n,\n…\n,\n1\n/\n\nζ\nn\n\n\n, and finally, the zeros of \n\n\np\n†\n\n\nz\n\n\n are \n\n1\n/\n\n\nζ\n1\n\n¯\n\n,\n…\n,\n1\n/\n\n\nζ\nn\n\n¯\n\n\n. Thus, if \n\np\n\nz\n\n\n has \n\nk\n\n zeros on \n\nR\n\n, \n\nl\n\n zeros on the upper half-plane \n\n\nC\n+\n\n=\n\n\nz\n∈\nC\n:\nIm\n\nz\n\n>\n0\n\n\n\n, and \n\nm\n\n zeros in the lower half-plane \n\n\nC\n−\n\n=\n\n\nz\n∈\nC\n:\nIm\n\nz\n\n<\n0\n\n\n\n so that \n\nk\n+\nl\n+\nm\n=\nn\n\n, then \n\n\np\n¯\n\n\nz\n\n\n will have the same number \n\nk\n\n of zeros on \n\nR\n\n, \n\nl\n\n zeros in \n\n\nC\n−\n\n\n and \n\nm\n\n zeros in \n\n\nC\n+\n\n\n. Similarly, if \n\np\n\nz\n\n\n has \n\nk\n\n zeros on \n\nS\n\n, \n\nl\n\n zeros inside \n\nS\n\n and \n\nm\n\n zeros outside \n\nS\n\n, so that \n\nk\n+\nl\n+\nm\n=\nn\n\n, then both \n\n\np\n∗\n\n\nz\n\n\n as \n\n\np\n†\n\n\nz\n\n\n will have the same number \n\nk\n\n of zeros on \n\nS\n\n, \n\nl\n\n zeros outside \n\nS\n\n and \n\nm\n\n zeros inside \n\nS\n\n.
\n
These properties encourage us to introduce the following classes of polynomials:
\n
Definition 2. A complex polynomial \n\np\n\nz\n\n\n is called1self-conjugate (SC), self-reciprocal (SR), or self-inversive (SI) if, for any zero \n\nζ\n\n of \n\np\n\nz\n\n\n, the complex-conjugate \n\n\nζ\n¯\n\n\n, the reciprocal \n\n1\n/\nζ\n\n, or the reciprocal of the complex-conjugate \n\n1\n/\n\nζ\n¯\n\n\n is also a zero of \n\np\n\nz\n\n\n, respectively.
\n
Thus, the zeros of any SC polynomial are all symmetric to the real line \n\nR\n\n, while the zeros of the any SI polynomial are symmetric to the unit circle \n\nS\n\n. The zeros of any SR polynomial are obtained by an inversion with respect to the unit circle followed by a reflection in the real line. From this, we can establish the following:
\n
Theorem 1.If\n\np\n\nz\n\n\nis an SC polynomial of odd degree, then it necessarily has at least one zero on\n\nR\n\n. Similarly, if\n\np\n\nz\n\n\nis an SR or SI polynomial of odd degree, then it necessarily has at least one zero on\n\nS\n\n.
\n
Proof. From Definition 2 it follows that the number of non-real zeros of an SC polynomial \n\np\n\nz\n\n\n can only occur in (conjugate) pairs; thus, if \n\np\n\nz\n\n\n has odd degree, then at least one zero of it must be real. Similarly, the zeros of \n\n\np\n†\n\n\nz\n\n\n or \n\n\np\n∗\n\n\nz\n\n\n that have modulus different from \n\n1\n\n can only occur in (inversive or reciprocal) pairs as well; thus, if \n\np\n\nz\n\n\n has odd degree then at least one zero of it must lie on \n\nS\n\n.□
\n
Theorem 2.The necessary and sufficient condition for a complex polynomial\n\np\n\nz\n\n\nto be SC, SR, or SI is that there exists a complex number\n\nω\n\nof modulus\n\n1\n\nso that one of the following relations, respectively, holds:
Proof. It is clear in view of (1) and (2) that these conditions are sufficient. We need to show, therefore, that these conditions are also necessary. Let us suppose first that \n\np\n\nz\n\n\n is SC. Then, for any zero \n\nζ\n\n of \n\np\n\nz\n\n\n the complex-conjugate number \n\n\nζ\n¯\n\n\n is also a zero of it. Thus, we can write
with \n\nω\n=\n\np\nn\n\n/\n\n\np\nn\n\n¯\n\n\n so that \n\n∣\nω\n∣\n=\n\n\n\np\nn\n\n/\n\n\np\nn\n\n¯\n\n\n\n=\n1\n\n. Now, let us suppose that \n\np\n\nz\n\n\n is SR. Then, for any zero \n\nζ\n\n of \n\np\n\nz\n\n\n, the reciprocal number \n\n1\n/\nζ\n\n is also a zero of it; thus,
with \n\nω\n=\n\n\n\n−\n1\n\n\nn\n\n/\n\n\n\nζ\n1\n\n…\n\nζ\nn\n\n\n\n=\n\np\nn\n\n/\n\np\n0\n\n\n; now, for any zero \n\nζ\n\n of \n\np\n\nz\n\n\n (which is necessarily different from zero if \n\np\n\nz\n\n\n is SR), there will be another zero whose value is \n\n1\n/\nζ\n\n so that \n\n\n\n\nζ\n1\n\n…\n\nζ\nn\n\n\n\n=\n1\n\n, which implies \n\n∣\nω\n∣\n=\n1\n\n. The proof for SI polynomials is analogous and will be concealed; it follows that \n\nω\n=\n\np\nn\n\n/\n\n\np\n0\n\n¯\n\n\n in this case.□
\n
Now from (1), (2) and (3), we can conclude that the coefficients of an SC, an SR, and an SI polynomial of degree \n\nn\n\n satisfy, respectively, the following relations:
We highlight that any real polynomial is SC—in fact, many theorems which are valid for real polynomials are also valid for, or can be easily extended to, SC polynomials.
\n
There also exist polynomials whose zeros are symmetric with respect to both the real line \n\nR\n\n and the unit circle \n\nS\n\n. A polynomial \n\np\n\nz\n\n\n with this double symmetry is, at the same time, SC and SI (and, hence, SR as well). This is only possible if all the coefficients of \n\np\n\nz\n\n\n are real, which implies that \n\nω\n=\n±\n1\n\n. This suggests the following additional definitions:
\n
Definition 3. A real self-reciprocal polynomial \n\np\n\nz\n\n\n that satisfies the relation \n\np\n\nz\n\n=\nω\n\nz\nn\n\np\n\n\n1\n/\nz\n\n\n\n will be called a positive self-reciprocal (PSR) polynomial if \n\nω\n=\n1\n\n and a negative self-reciprocal (NSR) polynomial if \n\nω\n=\n−\n1\n\n.
\n
Thus, the coefficients of any PSR polynomial \n\np\n\nz\n\n=\n\np\n0\n\n+\n⋯\n+\n\np\nn\n\n\nz\nn\n\n\n of degree \n\nn\n\n satisfy the relations \n\n\np\nk\n\n=\n\np\n\nn\n−\nk\n\n\n\n for \n\n0\n⩽\nk\n⩽\nn\n\n, while the coefficients of any NSR polynomial \n\np\n\nz\n\n\n of degree \n\nn\n\n satisfy the relations \n\n\np\nk\n\n=\n−\n\np\n\nn\n−\nk\n\n\n\n for \n\n0\n⩽\nk\n⩽\nn\n\n; this last condition implies that the middle coefficient of an NSR polynomial of even degree is always zero.
\n
Some elementary properties of PSR and NSR polynomials are the following: first, notice that, if \n\nζ\n\n is a zero of any PSR or NSR polynomial \n\np\n\nz\n\n\n of degree \n\nn\n⩾\n4\n\n, then the three complex numbers \n\n1\n/\nζ\n\n, \n\n\nζ\n¯\n\n\n and \n\n1\n/\n\nζ\n¯\n\n\n are also zeros of \n\np\n\nz\n\n\n. In particular, the number of zeros of such polynomials which are neither in \n\nS\n\n or in \n\nR\n\n is always a multiple of \n\n4\n\n. Besides, any NSR polynomial has \n\nz\n=\n1\n\n as a zero and \n\np\n\nz\n\n/\n\n\nz\n−\n1\n\n\n\n is PSR; further, if \n\np\n\nz\n\n\n has even degree then \n\nz\n=\n−\n1\n\n is also a zero of it and \n\np\n\nz\n\n/\n\n\n\nz\n2\n\n−\n1\n\n\n\n is a PSR polynomial of even degree. In a similar way, any PSR polynomial \n\np\n\nz\n\n\n of odd degree has \n\nz\n=\n−\n1\n\n as a zero and \n\np\n\nz\n\n/\n\n\nz\n+\n1\n\n\n\n is also PSR. The product of two PSR, or two NSR, polynomials is PSR, while the product of a PSR polynomial with an NSR polynomial is NSR. These statements follow directly from the definitions of such polynomials.
\n
We also mention that any PSR polynomial of even degree (say, \n\nn\n=\n2\nm\n\n) can be written in the following form:
an expression that is obtained by using the relations \n\n\np\nk\n\n=\n\np\n\n2\nm\n−\nk\n\n\n\n, \n\n0\n⩽\nk\n⩽\n2\nm\n\n, and gathering the terms of \n\np\n\nz\n\n\n with the same coefficients. Furthermore, the expression \n\n\nZ\ns\n\n\nz\n\n=\n\n\n\nz\ns\n\n+\n\nz\n\n−\ns\n\n\n\n\n\n for any integer \n\ns\n\n can be written as a polynomial of degree \n\ns\n\n in the new variable \n\nx\n=\nz\n+\n1\n/\nz\n\n (the proof follows easily by induction over \n\ns\n\n); thus, we can write \n\np\n\nz\n\n=\n\nz\nm\n\nq\n\nx\n\n\n, where \n\nq\n\nx\n\n=\n\nq\n0\n\n+\n⋯\n+\n\nq\nm\n\n\nx\nm\n\n\n is such that the coefficients \n\n\nq\n0\n\n,\n…\n,\n\nq\nm\n\n\n are certain functions of \n\n\np\n0\n\n,\n…\n,\n\np\nm\n\n\n. From this we can state the following:
\n
Theorem 3.Let\n\np\n\nz\n\n\nbe a PSR polynomial of even degree\n\nn\n=\n2\nm\n\n. For each pair\n\nζ\n\nand\n\n1\n/\nζ\n\nof self-reciprocal zeros of\n\np\n\nz\n\n\nthat lie on\n\nS\n\n, there is a corresponding zero\n\nξ\n\nof the polynomial\n\nq\n\nx\n\n\n, as defined above, in the interval\n\n\n\n−\n2\n\n2\n\n\nof the real line.
\n
Proof. For each zero \n\nζ\n\n of \n\np\n\nx\n\n\n that lie on \n\nS\n\n, write \n\nζ\n=\n\ne\niθ\n\n\n for some \n\nθ\n∈\nR\n\n. Thereby, as \n\nq\n\nx\n\n=\nq\n\n\nz\n+\n1\n/\nz\n\n\n=\np\n\nz\n\n/\n\nz\nm\n\n\n, it follows that \n\nξ\n=\nζ\n+\n1\n/\nζ\n=\n2\ncos\nθ\n\n will be a zero of \n\nq\n\nx\n\n\n. This shows us that \n\nξ\n\n is limited to the interval \n\n\n\n−\n2\n\n2\n\n\n of the real line. Finally, notice that the reciprocal zero \n\n1\n/\nζ\n\n of \n\np\n\nz\n\n\n is mapped to the same zero \n\nξ\n\n of \n\nq\n\nx\n\n\n.□
\n
Finally, remembering that the Chebyshev polynomials of first kind, \n\n\nT\nn\n\n\nz\n\n\n, are defined by the formula \n\n\nT\nn\n\n\n\n\n1\n2\n\n\n\nz\n+\n\nz\n\n−\n1\n\n\n\n\n\n\n=\n\n1\n2\n\n\n\n\nz\nn\n\n+\n\nz\n\n−\nn\n\n\n\n\n\n for \n\nz\n∈\nC\n\n, it follows as well that \n\nq\n\nx\n\n\n, and hence any PSR polynomial, can be written as a linear combination of Chebyshev polynomials:
3. How these polynomials are related to each other?
\n
In this section, we shall analyze how SC, SR, and SI polynomials are related to each other. Let us begin with the relationship between the SR and SI polynomials, which is actually very simple: indeed, from (1), (2), and (3) we can see that each one is nothing but the conjugate polynomial of the other, that is
Thus, if \n\np\n\nz\n\n\n is an SR (SI) polynomial, then \n\n\np\n¯\n\n\nz\n\n\n will be SI (SR) polynomial. Because of this simple relationship, several theorems which are valid for SI polynomials are also valid for SR polynomials and vice versa.
\n
The relationship between SC and SI polynomials is not so easy to perceive. A way of revealing their connection is to make use of a suitable pair of Möbius transformations, that maps the unit circle onto the real line and vice versa, which is often called Cayley transformations, defined through the formulas:
This approach was developed in [4], where some algorithms for counting the number of zeros that a complex polynomial has on the unit circle were also formulated.
\n
It is an easy matter to verify that \n\nM\n\nz\n\n\n maps \n\nR\n\n onto \n\nS\n\n while \n\nW\n\nz\n\n\n maps \n\nS\n\n onto \n\nR\n\n. Besides, \n\nM\n\nz\n\n\n maps the upper (lower) half-plane to the interior (exterior) of \n\nS\n\n, while \n\nW\n\nz\n\n\n maps the interior (exterior) of \n\nS\n\n onto the upper (lower) half-plane. Notice that \n\nW\n\nz\n\n\n can be thought as the inverse of \n\nM\n\nz\n\n\n in the Riemann sphere \n\n\nC\n∞\n\n=\nC\n∪\n\n∞\n\n\n, if we further assume that \n\nM\n\n\n−\ni\n\n\n=\n∞\n\n, \n\nM\n\n∞\n\n=\n1\n\n, \n\nW\n\n1\n\n=\n∞\n\n, and \n\nW\n\n∞\n\n=\n−\ni\n\n.
\n
Given a polynomial \n\np\n\nz\n\n\n of degree \n\nn\n\n, we define two Möbius-transformed polynomials, namely
The following theorem shows us how the zeros of \n\nQ\n\nz\n\n\n and \n\nT\n\nz\n\n\n are related with the zeros of \n\np\n\nz\n\n\n:
\n
Theorem 4.Let\n\n\nζ\n1\n\n,\n…\n,\n\nζ\nn\n\n\ndenote the zeros of\n\np\n\nz\n\n\nand\n\n\nη\n1\n\n,\n…\n,\n\nη\nn\n\n\nthe respective zeros of\n\nQ\n\nz\n\n\n. Provided\n\np\n\n1\n\n≠\n0\n\n, we have that\n\n\nη\n1\n\n=\nW\n\n\nζ\n1\n\n\n,\n…\n,\n\nη\nn\n\n=\nW\n\n\nζ\nn\n\n\n\n. Similarly, if\n\n\nτ\n1\n\n,\n…\n\nτ\nn\n\n\nare the zeros of\n\nT\n\nz\n\n\n, then we have\n\n\nτ\n1\n\n=\nM\n\n\nζ\n1\n\n\n,\n…\n,\n\nτ\nn\n\n=\nM\n\n\nζ\nn\n\n\n\n, provided that\n\np\n\n\n−\ni\n\n\n≠\n0\n\n.
\n
Proof. In fact, inverting the expression for \n\nQ\n\nz\n\n\n and evaluating it in any zero \n\n\nζ\nk\n\n\n of \n\np\n\nz\n\n\n we get that \n\np\n\n\nζ\nk\n\n\n=\n\n\n\n−\ni\n/\n2\n\n\nn\n\n\n\n\n\nζ\nk\n\n−\n1\n\n\nn\n\nQ\n\n\nW\n\n\nζ\nk\n\n\n\n\n=\n0\n\n for \n\n0\n⩽\nk\n\n⩽\nn\n\n. Provided that \n\nz\n=\n1\n\n is not a zero of \n\np\n\nz\n\n\n we get that \n\n\nη\nk\n\n=\nW\n\n\nζ\nk\n\n\n\n is a zero of \n\nQ\n\nz\n\n\n. The proof for the zeros of \n\nT\n\nz\n\n\n is analogous.□
\n
This result also shows that \n\nQ\n\nz\n\n\n and \n\nT\n\nz\n\n\n have the same degree as \n\np\n\nz\n\n\n whenever \n\np\n\n1\n\n≠\n0\n\n or \n\np\n\n\n−\ni\n\n\n≠\n0\n\n, respectively. In fact, if \n\np\n\nz\n\n\n has a zero at \n\nz\n=\n1\n\n of multiplicity \n\nm\n\n then \n\nQ\n\nz\n\n\n will be a polynomial of degree \n\nn\n−\nm\n\n, the same being true for \n\nT\n\nz\n\n\n if \n\np\n\nz\n\n\n has a zero of multiplicity \n\nm\n\n at \n\nz\n=\n−\ni\n\n. This can be explained by the fact that the points \n\nz\n=\n1\n\n and \n\nz\n=\n−\ni\n\n are mapped to infinity by \n\nW\n\nz\n\n\n and \n\nM\n\nz\n\n\n, respectively.
\n
The following theorem shows that the set of SI polynomials are isomorphic to the set of SC polynomials:
\n
Theorem 5.Let\n\np\n\nz\n\n\nbe an SI polynomial. Then, the transformed polynomial\n\nQ\n\nz\n\n=\n\n\n\nz\n+\ni\n\n\nn\n\np\n\n\nM\n\nz\n\n\n\n\nis an SC polynomial. Similarly, if\n\np\n\nz\n\n\nis an SC polynomial, then\n\nT\n\nz\n\n=\n\n\n\nz\n−\n1\n\n\nn\n\np\n\n\nW\n\nz\n\n\n\n\nwill be an SI polynomial.
\n
Proof. Let \n\nζ\n\n and \n\n1\n/\n\nζ\n¯\n\n\n be two inversive zeros an SI polynomial \n\np\n\nz\n\n\n. Then, according to Theorem 4, the corresponding zeros of \n\nQ\n\nz\n\n\n will be:
Thus, any pair of zeros of \n\np\n\nz\n\n\n that are symmetric to the unit circle are mapped in zeros of \n\nQ\n\nz\n\n\n that are symmetric to the real line; because \n\np\n\nz\n\n\n is SI, it follows that \n\nQ\n\nz\n\n\n is SC. Conversely, let \n\nζ\n\n and \n\n\nζ\n¯\n\n\n be two zeros of an SC polynomial \n\np\n\nz\n\n\n; then the corresponding zeros of \n\nT\n\nz\n\n\n will be:
Thus, any pair of zeros of \n\np\n\nz\n\n\n that are symmetric to the real line are mapped in zeros of \n\nT\n\nz\n\n\n that are symmetric to the unit circle. Because \n\np\n\nz\n\n\n is SC, it follows that \n\nT\n\nz\n\n\n is SI.□
\n
We can also verify that any SI polynomial with \n\nω\n=\n1\n\n is mapped to a real polynomial through \n\nM\n\nz\n\n\n and any real polynomial is mapped to an SI polynomial with \n\nω\n=\n1\n\n through \n\nW\n\nz\n\n\n. Thus, the set of SI polynomials with \n\nω\n=\n1\n\n is isomorphic to the set of real polynomials. Besides, an SI polynomial with \n\nω\n≠\n1\n\n can be transformed into another one with \n\nω\n=\n1\n\n by performing a suitable uniform rotation of its zeros. It can also be shown that the action of the Möbius transformation over a PSR polynomial leads to a real polynomial that has only even powers. See [4] for more.
\n
\n
\n
4. Zeros location theorems
\n
In this section, we shall discuss some theorems regarding the distribution of the zeros of SC, SR, and SI polynomials on the complex plane. Some general theorems relying on the number of zeros that an arbitrary complex polynomial has inside, on, or outside \n\nS\n\n are also discussed. To save space, we shall not present the proofs of these theorems, which can be found in the original works. Other related theorems can be found in Marden’s book [1].
\n
\n
4.1 Polynomials that do not necessarily have symmetric zeros
\n
The following theorems are classics (see [1] for the proofs):
\n
Theorem 6. (Rouché). Let\n\nq\n\nz\n\n\nand\n\nr\n\nz\n\n\nbe polynomials such that\n\n∣\nq\n\nz\n\n∣\n<\n∣\nr\n\nz\n\n∣\n\nalong all points of\n\nS\n\n. Then, the polynomial\n\np\n\nz\n\n=\nq\n\nz\n\n+\nr\n\nz\n\n\nhas the same number of zeros inside\n\nS\n\nas the polynomial\n\nr\n\nz\n\n\n, counted with multiplicity.
\n
Thus, if a complex polynomial \n\np\n\nz\n\n=\n\np\n0\n\n+\n⋯\n+\n\np\nk\n\n\nz\nk\n\n+\n⋯\n+\n\np\nn\n\n\nz\nn\n\n\n of degree \n\nn\n\n is such that \n\n\n\np\nk\n\n\n>\n\n\n\np\n0\n\n+\n⋯\n+\n\np\n\nk\n−\n1\n\n\n+\n\np\n\nk\n+\n1\n\n\n+\n⋯\n+\n\np\nn\n\n\n\n\n, then \n\np\n\nz\n\n\n will have exactly \n\nk\n\n zeros inside \n\nS\n\n, counted with multiplicity.
\n
Theorem 7. (Gauss and Lucas)The zeros of the derivative\n\n\np\n′\n\n\nz\n\n\nof a polynomial\n\np\n\nz\n\n\nlie all within the convex hull of the zeros of the\n\np\n\nz\n\n\n.
\n
Thereby, if a polynomial \n\np\n\nz\n\n\n has all its zeros on \n\nS\n\n, then all the zeros of \n\n\np\n′\n\n\nz\n\n\n will lie in or on \n\nS\n\n. In particular, the zeros of \n\n\np\n′\n\n\nz\n\n\n will lie on \n\nS\n\n if, and only if, they are multiple zeros of \n\np\n\nz\n\n\n.
\n
Theorem 8. (Cohn)A necessary and sufficient condition for all the zeros of a complex polynomial\n\np\n\nz\n\n\nto lie on\n\nS\n\nis that\n\np\n\nz\n\n\nis SI and that its derivative\n\n\np\n′\n\n\nz\n\n\ndoes not have any zero outside\n\nS\n\n.
\n
Cohn introduced his theorem in [5]. Bonsall and Marden presented a simpler proof of Conh’s theorem in [6] (see also [7]) and applied it to SI polynomials—in fact, this was probably the first paper to use the expression “self-inversive.” Other important result of Cohn is the following: all the zeros of a complex polynomial \n\np\n\nz\n\n=\n\np\nn\n\n\nz\nn\n\n+\n⋯\n+\n\np\n0\n\n\n will lie on \n\nS\n\n if, and only if, \n\n∣\n\np\nn\n\n∣\n=\n∣\n\np\n0\n\n∣\n\n and all the zeros of \n\np\n\nz\n\n\n do not lie outside \n\nS\n\n.
\n
Restricting ourselves to polynomials with real coefficients, Eneström and Kakeya [8, 9, 10] independently presented the following theorem:
\n
Theorem 9. (Eneström and Kakeya)Let\n\np\n\nz\n\n\nbe a polynomial of degree\n\nn\n\nwith real coefficients. If its coefficients are such that\n\n0\n<\n\np\n0\n\n⩽\n\np\n1\n\n⩽\n⋯\n⩽\n\np\n\nn\n−\n1\n\n\n⩽\n\np\nn\n\n\n, then all the zeros of\n\np\n\nz\n\n\nlie in or on\n\nS\n\n. Likewise, if the coefficients of\n\np\n\nz\n\n\nare such that\n\n0\n<\n\np\nn\n\n⩽\n\np\n\nn\n−\n1\n\n\n⩽\n⋯\n⩽\n\np\n1\n\n⩽\n\np\n0\n\n\n, then all the zeros of\n\np\n\nz\n\n\nlie on or outside\n\nS\n\n.
\n
The following theorems are relatively more recent. The distribution of the zeros of a complex polynomial regarding the unit circle \n\nS\n\n was presented by Marden in [1] and slightly enhanced by Jury in [11]:
\n
Theorem 10. (Marden and Jury)Let\n\np\n\nz\n\n\nbe a complex polynomial of degree\n\nn\n\nand\n\n\np\n∗\n\n\nz\n\n\nits reciprocal. Construct the sequence of polynomials\n\n\nP\nj\n\n\nz\n\n=\n\n∑\n\nk\n=\n0\n\n\nn\n−\nj\n\n\n\nP\n\nj\n,\nk\n\n\n\nz\nk\n\n\nsuch that\n\n\nP\n0\n\n\nz\n\n=\np\n\nz\n\n\nand\n\n\nP\n\nj\n+\n1\n\n\n\nz\n\n=\n\n\np\n\nj\n,\n0\n\n\n¯\n\n\nP\nj\n\n\nz\n\n−\n\n\np\n\nj\n,\nn\n−\nj\n\n\n¯\n\n\nP\nj\n∗\n\n\nz\n\n\nfor\n\n0\n⩽\nj\n⩽\nn\n−\n1\n\nso that we have the relations\n\n\np\n\nj\n+\n1\n,\nk\n\n\n=\n\n\np\n\nj\n,\n0\n\n\n¯\n\n\np\n\nj\n,\nk\n\n\n−\n\np\n\nj\n,\nn\n−\nj\n\n\n\n\np\n\nj\n,\nn\n−\nj\n−\nk\n\n\n¯\n\n\n. Let\n\n\nδ\nj\n\n\ndenote the constant terms of the polynomials\n\n\nP\nj\n\n\nz\n\n\n, i.e.,\n\n\nδ\nj\n\n=\n\np\n\nj\n,\n0\n\n\n\nand\n\n\nΔ\nk\n\n=\n\nδ\n1\n\n⋯\n\nδ\nk\n\n\n. Thus, if\n\nN\n\nof the products\n\n\nΔ\nk\n\n\nare negative and\n\nn\n−\nN\n\nof the products\n\n\nΔ\nk\n\n\nare positive so that none of them are zero, then\n\np\n\nz\n\n\nhas\n\nN\n\nzeros inside\n\nS\n\n,\n\nn\n−\nN\n\nzeros outside\n\nS\n\nand no zero on\n\nS\n\n. On the other hand, if\n\n\nΔ\nk\n\n≠\n0\n\nfor some\n\nk\n<\nn\n\nbut\n\n\nP\n\nk\n+\n1\n\n\n\nz\n\n=\n0\n\n, then\n\np\n\nz\n\n\nhas either\n\nn\n−\nk\n\nzeros on\n\nS\n\nor\n\nn\n−\nk\n\nzeros symmetric to\n\nS\n\n. It has additionally\n\nN\n\nzeros inside\n\nS\n\nand\n\nk\n−\nN\n\nzeros outside\n\nS\n\n.
\n
A simple necessary and sufficient condition for all the zeros of a complex polynomial to lie on \n\nS\n\n was presented by Chen in [12]:
\n
Theorem 11. (Chen)A necessary and sufficient condition for all the zeros of a complex polynomial\n\np\n\nz\n\n\nof degree\n\nn\n\nto lie on\n\nS\n\nis that there exists a polynomial\n\nq\n\nz\n\n\nof degree\n\nn\n−\nm\n\nwhose zeros are all in or on\n\nS\n\nand such that\n\np\n\nz\n\n=\n\nz\nm\n\nq\n\nz\n\n+\nω\n\nq\n†\n\n\nz\n\n\nfor some complex number\n\nω\n\nof modulus\n\n1\n\n.
\n
We close this section by mentioning that there exist many other well-known theorems regarding the distribution of the zeros of complex polynomials. We can cite, for example, the famous rule of Descartes (the number of positive zeros of a real polynomial is limited from above by the number of sign variations in the ordered sequence of its coefficients), the Sturm Theorem (the exact number of zeros that a real polynomial has in a given interval \n\n\na\nb\n\n\n of the real line is determined by the formula \n\nN\n=\nvar\n\n\nS\n\nb\n\n\n\n−\nvar\n\n\nS\n\na\n\n\n\n\n, where \n\nvar\n\n\nS\n\nξ\n\n\n\n\n means the number of sign variations of the Sturm sequence \n\nS\n\nx\n\n\n evaluated at \n\nx\n=\nξ\n\n) and Kronecker Theorem (if all the zeros of a monic polynomial with integer coefficients lie on the unit circle, then all these zeros are indeed roots of unity), see [1] for more. There are still other important theorems relying on matrix methods and quadratic forms that were developed by several authors as Cohn, Schur, Hermite, Sylvester, Hurwitz, Krein, among others, see [13].
\n
\n
\n
4.2 Real self-reciprocal polynomials
\n
Let us now consider real SR polynomials. The theorems below are usually applied to PSR polynomials, but some of them can be extended to NSR polynomials as well.
\n
An analog of Eneström-Kakeya theorem for PSR polynomials was found by Chen in [12] and then, in a slightly stronger version, by Chinen in [14]:
\n
Theorem 12. (Chen and Chinen)Let\n\np\n\nz\n\n\nbe a PSR polynomial of degree\n\nn\n\nthat is written in the form\n\np\n\nz\n\n=\n\np\n0\n\n+\n\np\n1\n\nz\n+\n⋯\n+\n\np\nk\n\n\nz\nk\n\n+\n\np\nk\n\n\nz\n\nn\n−\nk\n\n\n+\n\np\n\nk\n−\n1\n\n\n\nz\n\nn\n−\nk\n+\n1\n\n\n+\n⋯\n+\n\np\n0\n\n\nz\nn\n\n\nand such that\n\n0\n<\n\np\nk\n\n<\n\np\n\nk\n−\n1\n\n\n<\n⋯\n<\n\np\n1\n\n<\n\np\n0\n\n\n. Then all the zeros of\n\np\n\nz\n\n\nare on\n\nS\n\n.
\n
Going in the same direction, Choo found in [15] the following condition:
\n
Theorem 13. (Choo)Let\n\np\n\nz\n\n\nbe a PSR polynomial of degree\n\nn\n\nand such that its coefficients satisfy the following conditions:\n\n\nnp\nn\n\n⩾\n\n\nn\n−\n1\n\n\n\np\n\nn\n−\n1\n\n\n⩾\n⋯\n⩾\n\n\nk\n+\n1\n\n\n\np\n\nk\n+\n1\n\n\n>\n0\n\nand\n\n\n\nk\n+\n1\n\n\n\np\n\nk\n+\n1\n\n\n\n⩾\n\n\n∑\n\nj\n=\n0\n\nk\n\n\n\n\n\nj\n+\n1\n\n\n\np\n\nj\n+\n1\n\n\n−\n\njp\nj\n\n\n\n\nfor\n\n0\n⩽\nk\n⩽\nn\n−\n1\n\n. Then, all the zeros of\n\np\n\nz\n\n\nare on\n\nS\n\n.
\n
Lakatos discussed the separation of the zeros on the unit circle of PSR polynomials in [16]; she also found several sufficient conditions for their zeros to be all on \n\nS\n\n. One of the main theorems is the following:
\n
Theorem 14. (Lakatos)Let\n\np\n\nz\n\n\nbe a PSR polynomial of degree\n\nn\n>\n2\n\n. If\n\n\n\np\nn\n\n\n⩾\n\n∑\n\nk\n=\n1\n\n\nn\n−\n1\n\n\n\n\n\np\nn\n\n−\n\np\nk\n\n\n\n\n, then all the zeros of\n\np\n\nz\n\n\nlie on\n\nS\n\n. Moreover, the zeros of\n\np\n\nz\n\n\nare all simple, except when the equality takes place.
\n
For PSR polynomials of odd degree, Lakatos and Losonczi [17] found a stronger version of this result:
\n
Theorem 15. (Lakatos and Losonczi)Let\n\np\n\nz\n\n\nbe a PSR polynomial of odd degree, say\n\nn\n=\n2\nm\n+\n1\n\n. If\n\n\n\np\n\n2\nm\n+\n1\n\n\n\n⩾\n\ncos\n2\n\n\n\nϕ\nm\n\n\n\n∑\n\nk\n=\n1\n\n\n2\nm\n\n\n\n\n\np\n\n2\nm\n+\n1\n\n\n−\n\np\nk\n\n\n\n\n, where\n\n\nϕ\nm\n\n=\nπ\n/\n\n\n4\n\n\nm\n+\n1\n\n\n\n\n\n, then all the zeros of\n\np\n\nz\n\n\nlie on\n\nS\n\n. The zeros are simple except when the equality is strict.
\n
Theorem 14 was generalized further by Lakatos and Losonczi in [18]:
\n
Theorem 16. (Lakatos and Losonczi)All zeros of a PSR polynomial\n\np\n\nz\n\n\nof degree\n\nn\n>\n2\n\nlie on\n\nS\n\nif the following conditions hold:\n\n\n\n\np\nn\n\n+\nr\n\n\n⩾\n\n∑\n\nk\n=\n1\n\n\nn\n−\n1\n\n\n\n\n\np\nk\n\n−\n\np\nn\n\n+\nr\n\n\n\n,\n\n\np\nn\n\nr\n⩾\n0\n\n, and\n\n\n\np\nn\n\n\n⩾\n\nr\n\n\n, for\n\nr\n∈\nR\n\n.
\n
Other conditions for all the zeros of a PSR polynomial to lie on \n\nS\n\n were presented by Kwon in [19]. In its simplest form, Kown’s theorem can be enunciated as follows:
\n
Theorem 17. (Kwon)Let\n\np\n\nz\n\n\nbe a PSR polynomial of even degree\n\nn\n⩾\n2\n\nwhose leading coefficient\n\n\np\nn\n\n\nis positive and\n\n\np\n0\n\n⩽\n\np\n1\n\n⩽\n⋯\n⩽\n\np\nn\n\n\n. In this case, all the zeros of\n\np\n\nz\n\n\nwill lie on\n\nS\n\nif, either\n\n\np\n\nn\n/\n2\n\n\n⩾\n\n∑\n\nk\n=\n0\n\nn\n\n\n\n\np\nk\n\n−\n\np\n\nn\n/\n2\n\n\n\n\n\n, or\n\np\n\n1\n\n⩾\n0\n\nand\n\n\np\nn\n\n⩾\n\n1\n2\n\n\n∑\n\nk\n=\n1\n\n\nn\n−\n1\n\n\n\n\n\np\nk\n\n−\n\np\n\nn\n/\n2\n\n\n\n\n\n.
\n
Modified forms of this theorem hold for PSR polynomials of odd degree and for the case where the coefficients of \n\np\n\nz\n\n\n do not have the ordination above—see [19] for these cases. Kwon also found conditions for all but two zeros of \n\np\n\nz\n\n\n to lie on \n\nS\n\n in [20], which is relevant to the theory of Salem polynomials—see Section 5.
\n
Other interesting results are the following: Konvalina and Matache [21] found conditions under which a PSR polynomial has at least one non-real zero on \n\nS\n\n. Kim and Park [22] and then Kim and Lee [23] presented conditions for which all the zeros of certain PSR polynomials lie on \n\nS\n\n (some open cases were also addressed by Botta et al. in [24]). Suzuki [25] presented necessary and sufficient conditions, relying on matrix algebra and differential equations, for all the zeros of PSR polynomials to lie on \n\nS\n\n. In [26] Botta et al. studied the distribution of the zeros of PSR polynomials with a small perturbation in their coefficients. Real SR polynomials of height \n\n1\n\n—namely, special cases of Littlewood, Newman, and Borwein polynomials—were studied by several authors, see [27, 28, 29, 30, 31, 32, 33, 34, 35] and references therein.2 Zeros of the so-called Ramanujan Polynomials and generalizations were analyzed in [37, 38, 39]. Finally, the Galois theory of PSR polynomials was studied in [40] by Lindstrøm, who showed that any PSR polynomial of degree less than \n\n10\n\n can be solved by radicals.
\n
\n
\n
4.3 Complex self-reciprocal and self-inversive polynomials
\n
Let us consider now the case of complex SR polynomials and SI polynomials. Here, we remark that many of the theorems that hold for SI polynomials either also hold for SR polynomials or can be easily adapted to this case (the opposite is also true).
\n
Theorem 18. (Cohn)An SI polynomial\n\np\n\nz\n\n\nhas as many zeros outside\n\nS\n\nas does its derivative\n\n\np\n′\n\n\nz\n\n\n.
\n
This follows directly from Cohn’s Theorem 8 for the case where \n\np\n\nz\n\n\n is SI. Besides, we can also conclude from this that the derivative of \n\np\n\nz\n\n\n has no zeros on \n\nS\n\n except at the multiple zeros of \n\np\n\nz\n\n\n. Furthermore, if an SI polynomial \n\np\n\nz\n\n\n of degree \n\nn\n\n has exactly \n\nk\n\n zeros on \n\nS\n\n, while its derivative has exactly \n\nl\n\n zeros in or on \n\nS\n\n, both counted with multiplicity, then \n\nn\n=\n2\n\n\nl\n+\n1\n\n\n−\nk\n\n.
\n
O’Hara and Rodriguez [41] showed that the following conditions are always satisfied by SI polynomials whose zeros are all on \n\nS\n\n:
\n
Theorem 19. (O’Hara and Rodriguez)Let\n\np\n\nz\n\n\nbe an SI polynomial of degree\n\nn\n\nwhose zeros are all on\n\nS\n\n. Then, the following inequality holds:\n\n\n∑\n\nj\n=\n0\n\nn\n\n\n\n\np\nj\n\n\n2\n\n⩽\n\n\n\np\n\nz\n\n\n\n2\n\n\n, where\n\n\n\np\n\nz\n\n\n\n\ndenotes the maximum modulus of\n\np\n\nz\n\n\non the unit circle; besides, if this inequality is strict then the zeros of\n\np\n\nz\n\n\nare rotations of\n\nn\n\nth roots of unity. Moreover, the following inequalities are also satisfied:\n\n\n\na\nk\n\n\n⩽\n\n1\n2\n\n\n\np\n\nz\n\n\n\n\n if \n\nk\n≠\nn\n/\n2\n\n and \n\n\n\na\nk\n\n\n⩽\n\n\n2\n\n2\n\n\n\np\n\nz\n\n\n\n\n for \n\nk\n=\nn\n/\n2\n\n.
\n
Schinzel in [42], generalized Lakatos Theorem 14 for SI polynomials:
\n
Theorem 20. (Schinzel)Let\n\np\n\nz\n\n\nbe an SI polynomial of degree\n\nn\n\n. If the inequality\n\n\n\np\nn\n\n\n⩾\n\ninf\n\na\n,\nb\n∈\nC\n:\n∣\nb\n∣\n=\n1\n\n\n\n∑\n\nk\n=\n0\n\nn\n\n\n\n\nap\nk\n\n−\n\nb\n\nn\n−\nk\n\n\n\np\nn\n\n\n\n\n, then all the zeros of\n\np\n\nz\n\n\nlie on\n\nS\n\n. These zero are simple whenever the equality is strict.
\n
In a similar way, Losonczi and Schinzel [43] generalized theorem 15 for the SI case:
\n
Theorem 21. (Losonczi and Schinzel)Let\n\np\n\nz\n\n\nbe an SI polynomial of odd degree, i.e.,\n\nn\n=\n2\nm\n+\n1\n\n. If\n\n\n\np\n\n2\nm\n+\n1\n\n\n\n⩾\n\ncos\n2\n\n\n\nϕ\nm\n\n\n\ninf\n\na\n,\nb\n∈\nC\n:\n∣\nb\n∣\n=\n1\n\n\n\n∑\n\nk\n=\n1\n\n\n2\nm\n+\n1\n\n\n\n\n\nap\nk\n\n−\n\nb\n\n2\nm\n+\n1\n−\nk\n\n\n\np\n\n2\nm\n+\n1\n\n\n\n\n\n, where\n\n\nϕ\nm\n\n=\nπ\n/\n\n\n4\n\n\nm\n+\n1\n\n\n\n\n\n, then all the zeros of\n\np\n\nz\n\n\nlie on\n\nS\n\n. The zeros are simple except when the equality is strict.
\n
Another sufficient condition for all the zeros of an SI polynomial to lie on \n\nS\n\n was presented by Lakatos and Losonczi in [44]:
\n
Theorem 22. (Lakatos and Losonczi)Let\n\np\n\nz\n\n\nbe an SI polynomial of degree\n\nn\n\nand suppose that the inequality\n\n\n\np\nn\n\n\n⩾\n\n\n1\n2\n\n\n∑\n\nk\n=\n1\n\n\nn\n−\n1\n\n\n\n\np\nk\n\n\n\nholds. Then, all the zeros of\n\np\n\nz\n\n\nlie on\n\nS\n\n. Moreover, the zeros are all simple except when an equality takes place.
\n
In [45], Lakatos and Losonczi also formulated a theorem that contains as special cases many of the previous results:
\n
Theorem 23. (Lakatos and Losonczi)Let\n\np\n\nz\n\n=\n\np\n0\n\n+\n⋯\n+\n\np\nn\n\n\nz\nn\n\n\nbe an SI polynomial of degree\n\nn\n⩾\n2\n\nand\n\na\n\n,\n\nb\n\n, and\n\nc\n\nbe complex numbers such that\n\na\n≠\n0\n\n,\n\n∣\nb\n∣\n=\n1\n\n, and\n\nc\n/\n\np\nn\n\n∈\nR\n\n,\n\n0\n⩽\nc\n/\n\np\nn\n\n⩽\n1\n\n. If\n\n\n\n\np\nn\n\n+\nc\n\n\n⩾\n\n\n\nap\n0\n\n−\n\nb\nn\n\n\np\nn\n\n\n\n+\n\n∑\n\nk\n=\n1\n\n\nn\n−\n1\n\n\n\n\n\nap\nk\n\n−\n\nb\n\nn\n−\nk\n\n\n\n\nc\n−\n\np\nn\n\n\n\n\n\n+\n\n\n\nap\nn\n\n−\n\np\nn\n\n\n\n\n, then, all the zeros of\n\np\n\nz\n\n\nlie on\n\nS\n\n. Moreover, these zeros are simple if the inequality is strict.
\n
In [46], Losonczi presented the following necessary and sufficient conditions for all the zeros of a (complex) SR polynomial of even degree to lie on \n\nS\n\n:
\n
Theorem 24. (Losonczi)Let\n\np\n\nz\n\n\nbe a monic complex SR polynomial of even degree, say\n\nn\n=\n2\nm\n\n. Then, all the zeros of\n\np\n\nz\n\n\nwill lie on\n\nS\n\nif, and only if, there exist real numbers\n\n\nα\n1\n\n,\n…\n,\n\nα\n\n2\nm\n\n\n\n, all with moduli less than or equal to\n\n2\n\n, that satisfy the inequalities:\n\n\np\nk\n\n=\n\n\n\n−\n1\n\n\nk\n\n\n∑\n\nl\n=\n0\n\n\n\nk\n/\n2\n\n\n\n\n\n\n\nm\n−\nk\n+\n2\nl\n\n\n\n\nl\n\n\n\n\n\nσ\n\nk\n−\n2\nl\n\n\n2\nm\n\n\n\n\nα\n1\n\n…\n\nα\n\n2\nm\n\n\n\n\n,\n\n0\n⩽\nk\n⩽\nm\n\n, where\n\n\nσ\nk\n\n2\nm\n\n\n\n\nα\n1\n\n…\n\nα\n\n2\nm\n\n\n\n\ndenotes the\n\nk\n\nth elementary symmetric function in the\n\n2\nm\n\nvariables\n\n\nα\n1\n\n,\n…\n,\n\nα\n\n2\nm\n\n\n\n.
\n
Losonczi, in [46], also showed that if all the zeros of a complex monic reciprocal polynomial are on \n\nS\n\n, then its coefficients are all real and satisfy the inequality \n\n\n\np\nn\n\n\n⩽\n\n\n\n\nn\n\n\n\n\nk\n\n\n\n\n\n for \n\n0\n⩽\nk\n⩽\nn\n\n.
\n
The theorems above give conditions for all the zeros of SI or SR polynomials to lie on \n\nS\n\n. In many cases, however, we need to verify if a polynomial has a given number of zeros (or none) on the unit circle. Considering this problem, Vieira in [47] found sufficient conditions for an SI polynomial of degree \n\nn\n\n to have a determined number of zeros on the unit circle. In terms of the length, \n\nL\n\n\np\n\nz\n\n\n\n=\n\n\np\n0\n\n\n+\n⋯\n+\n\n\np\nn\n\n\n\n of a polynomial \n\np\n\nz\n\n\n of degree \n\nn\n\n, this theorem can be stated as follows:
\n
Theorem 25. (Vieira)Let\n\np\n\nz\n\n\nbe an SI polynomial of degree\n\nn\n\n. If the inequality\n\n\n\np\n\nn\n−\nm\n\n\n\n⩾\n\n1\n4\n\n\n\nn\n\nn\n−\nm\n\n\n\nL\n\n\np\n\nz\n\n\n\n\n,\n\nm\n<\nn\n/\n2\n\n, holds true, then\n\np\n\nz\n\n\nwill have exactly\n\nn\n−\n2\nm\n\nzeros on\n\nS\n\n; besides, all these zeros are simple when the inequality is strict. Moreover,\n\np\n\nz\n\n\nwill have no zero on\n\nS\n\nif, for\n\nn\n\neven and\n\nm\n=\nn\n/\n2\n\n, the inequality\n\n\n\np\nm\n\n\n>\n\n1\n2\n\nL\n\n\np\n\nz\n\n\n\n\nis satisfied.
\n
The case \n\nm\n=\n0\n\n corresponds to Lakatos and Losonczi Theorem 14 for all the zeros of \n\np\n\nz\n\n\n to lie on \n\nS\n\n. The necessary counterpart of this theorem was considered by Stankov in [48], with an application to the theory of Salem numbers—see Section 5.1.
\n
Other results on the distribution of zeros of SI polynomials include the following: Sinclair and Vaaler [49] showed that a monic SI polynomial \n\np\n\nz\n\n\n of degree \n\nn\n\n satisfying the inequalities \n\n\nL\nr\n\n\n\np\n\nz\n\n\n\n⩽\n2\n+\n\n2\nr\n\n\n\n\nn\n−\n1\n\n\n\n1\n−\nr\n\n\n\n or \n\n\nL\nr\n\n\n\np\n\nz\n\n\n\n⩽\n2\n+\n\n2\nr\n\n\n\n\nl\n−\n2\n\n\n\n1\n−\nr\n\n\n\n, where \n\nr\n⩾\n1\n\n, \n\n\nL\nr\n\n\n\np\n\nz\n\n\n\n=\n\n\n\np\n0\n\n\nr\n\n+\n⋯\n+\n\n\n\np\nn\n\n\nr\n\n\n, and \n\nl\n\n is the number of non-null terms of \n\np\n\nz\n\n\n, has all their zeros on \n\nS\n\n; the authors also studied the geometry of SI polynomials whose zeros are all on \n\nS\n\n. Choo and Kim applied Theorem 11 to SI polynomials in [50]. Hypergeometric polynomials with all their zeros on \n\nS\n\n were considered in [51, 52]. Kim [53] also obtained SI polynomials which are related to Jacobi polynomials. Ito and Wimmer [54] studied SI polynomial operators in Hilbert space whose spectrum is on \n\nS\n\n.
\n
\n
\n
\n
5. Where these polynomials are found?
\n
In this section, we shall briefly discuss some important or recent applications of the theory of polynomials with symmetric zeros. We remark, however, that our selection is by no means exhaustive: for example, SR and SI polynomials also find applications in many fields of mathematics (e.g., information and coding theory [55], algebraic curves over a finite field and cryptography [56], elliptic functions [57], number theory [58], etc.) and physics (e.g., Lee-Yang theorem in statistical physics [59], Poincaré Polynomials defined on Calabi-Yau manifolds of superstring theory [60], etc.).
\n
\n
5.1 Polynomials with small Mahler measure
\n
Given a monic polynomial \n\np\n\nz\n\n\n of degree \n\nn\n\n, with integer coefficients, the Mahler measure of \n\np\n\nz\n\n\n, denoted by \n\nM\n\n\np\n\nz\n\n\n\n\n, is defined as the product of the modulus of all those zeros of \n\np\n\nz\n\n\n that lie in the exterior of \n\nS\n\n [61]. That is
where \n\n\nζ\n1\n\n,\n…\n,\n\nζ\nn\n\n\n are the zeros3 of \n\np\n\nz\n\n\n. Thus, if a monic integer polynomial \n\np\n\nz\n\n\n has all its zeros in or on the unit circle, we have \n\nM\n\n\np\n\nz\n\n\n\n=\n1\n\n; in particular, all cyclotomic polynomials (which are PSR polynomials whose zeros are the primitive roots of unity, see [1]) have Mahler measure equal to \n\n1\n\n. In a sense, the Mahler measure of a polynomial \n\np\n\nz\n\n\n measures how close it is to the cyclotomic polynomials. Therefore, it is natural to raise the following:
\n
Problem 1. (Mahler)Find the monic, integer, non-cyclotomic polynomial with the smallest Mahler measure.
\n
This is an 80-year-old open problem of mathematics. Of course, we can expect that the polynomials with the smallest Mahler measure be among those with only a few number of zeros outside \n\nS\n\n, in particular among those with only one zero outside \n\nS\n\n. A monic integer polynomial that has exactly one zero outside \n\nS\n\n is called a Pisot polynomial and its unique zero of modulus greater than \n\n1\n\n is called its Pisot number [62]. A breakthrough towards the solution of Mahler’s problem was given by Smyth in [63]:
\n
Theorem 26. (Smyth)The Pisot polynomial\n\nS\n\nz\n\n=\n\nz\n3\n\n−\nz\n−\n1\n\nis the polynomial with smallest Mahler measure among the set of all monic, integer, and non-SR polynomials. Its Mahler measure is given by the value of its Pisot number, which is,
The Mahler problem is, however, still open for SR polynomials. A monic integer SR polynomial with exactly two (real and positive) zeros (say, \n\nζ\n\n and \n\n1\n/\nζ\n\n) not lying on \n\nS\n\n is called a Salem polynomial [62, 64]. It can be shown that a Pisot polynomial with at least one zero on \n\nS\n\n is also a Salem polynomial. The unique positive zero greater than one of a Salem polynomial is called its Salem number, which also equals the value of its Mahler measure. A Salem number \n\ns\n\n is said to be small if \n\ns\n<\nσ\n\n; up to date, only \n\n47\n\n small Salem numbers are known [65, 66] and the smallest known one was found about 80 years ago by Lehmer [67]. This gave place to the following:
\n
Conjecture 1. (Lehmer)The monic integer polynomial with the smallest Mahler measure is the Lehmer polynomial\n\nL\n\nz\n\n=\n\nz\n10\n\n+\n\nz\n9\n\n−\n\nz\n7\n\n−\n\nz\n6\n\n−\n\nz\n5\n\n−\n\nz\n4\n\n−\n\nz\n3\n\n+\nz\n+\n1\n\n, a Salem polynomial whose Mahler measure is\n\nΛ\n≈\n1.17628081826\n\n, known as Lehmer’s constant.
\n
The proof of this conjecture is also an open problem. To be fair, we do not even know if there exists a smallest Salem number at all. This is the content of another problem raised by Lehmer:
\n
Problem 2. (Lehmer)Answer whether there exists or not a positive number\n\nϵ\n\nsuch that the Mahler measure of any monic, integer, and non-cyclotomic polynomial\n\np\n\nz\n\n\nsatisfies the inequality\n\nM\n\n\np\n\nz\n\n\n\n>\n1\n+\nϵ\n\n.
\n
Lehmer’s polynomial also appears in connection with several fields of mathematics. Many examples are discussed in Hironaka’s paper [68]; here we shall only present an amazing identity found by Bailey and Broadhurst in [69] in their works on polylogarithm ladders: if \n\nλ\n\n is any zero of the aforementioned Lehmer’s polynomial \n\nL\n\nz\n\n\n, then,
A knot is a closed, non-intersecting, one-dimensional curve embedded on \n\n\nR\n3\n\n\n [70]. Knot theory studies topological properties of knots as, for example, criteria under which a knot can be unknot, conditions for the equivalency between knots, the classification of prime knots, etc.; see [70] for the corresponding definitions. In Figure 1, we plotted all prime knots up to six crossings.
\n
Figure 1.
A table of prime knots up to six crossings. In the Alexander-Briggs notation these knots are, in order, \n\n\n0\n1\n\n\n, \n\n\n3\n1\n\n\n, \n\n\n4\n1\n\n\n, \n\n\n5\n1\n\n\n, \n\n\n5\n2\n\n\n, \n\n\n6\n1\n\n\n, \n\n\n6\n2\n\n\n, and \n\n\n6\n3\n\n\n.
\n
One of the most important questions in knot theory is to determine whether or not two knots are equivalent. This, however, is not an easy task. A way of attacking this question is to look for abstract objects—mainly the so-called knot invariants—rather than to the knots themselves. A knot invariant is a (topologic, combinatorial, algebraic, etc.) quantity that can be computed for any knot and that is always the same for equivalents knots.4 An important class of knot invariants is constituted by the so-called Knot Polynomials. Knot polynomials were introduced in 1928 by Alexander [71]. They consist in polynomials with integer coefficients that can be written down for every knot. For about 60 years since its creation, Alexander polynomials were the only known kind of knot polynomial. It was only in 1985 that Jones [72] came up with a new kind of knot polynomials—today known as Jones polynomials—and since then other kinds were discovered as well, see [70].
\n
What is interesting for us here is that the Alexander polynomials are PSR polynomials of even degree (say, \n\nn\n=\n2\nm\n\n) and with integer coefficients.5 Thus, they have the following general form:
where \n\n\nδ\ni\n\n∈\nN\n\n, \n\n0\n⩽\ni\n⩽\nm\n\n. In Table 1, we present the $\\delta_{m - 1}$Alexander polynomials for the prime knots up to six crossings.
Alexander polynomials for prime knots up to six crossings.
\n
Knots theory finds applications in many fields of mathematics in physics—see [70]. In mathematics, we can cite a very interesting connection between Alexander polynomials and the theory of Salem numbers: more precisely, the Alexander polynomial associated with the so-called Pretzel Knot\n\nP\n\n\n−\n2,3,7\n\n\n\n is nothing but the Lehmer polynomial \n\nL\n\nz\n\n\n introduced in Section 5.1; it is indeed the Alexander polynomial with the smallest Mahler measure [73]. In physics, knot theory is connected with quantum groups and it also can be used to one construct solutions of the Yang-Baxter equation [74] through a method called baxterization of braid groups.
\n
\n
\n
5.3 Bethe equations
\n
Bethe equations were introduced in 1931 by Hans Bethe [75], together with his powerful method—the so-called Bethe Ansatz Method—for solving spectral problems associated with exactly integrable models of statistical mechanics. They consist in a system of coupled and non-linear equations that ensure the consistency of the Bethe Ansatz. In fact, for the XXZ Heisenberg spin chain, the Bethe Equations consist in a coupled system of trigonometric equations; however, after a change of variables is performed, we can write them in the following rational form:
where L ∈ \n\nN\n\n is the length of the chain, N ∈ \n\nN\n\n is the excitation number and Δ ∈ \n\nR\n\n is the so-called spectral parameter. A solution of (18) consists in a (non-ordered) set \n\nX\n=\n\n\nx\n1\n\n…\n\nx\nN\n\n\n\n of the unknowns \n\n\nx\n1\n\n,\n…\n,\n\nx\nN\n\n\n so that (18) is satisfied. Notice that the Bethe equations satisfy the important relation \n\n\nx\n1\nL\n\n\nx\n2\nL\n\n⋯\n\nx\nN\nL\n\n=\n1\n\n, which suggests an inversive symmetry of their zeros.
\n
In [76], Vieira and Lima-Santos showed that the solutions of (18), for \n\nN\n=\n2\n\n and arbitrary \n\nL\n\n, are given in terms of the zeros of certain SI polynomials. In fact, (18) becomes a system of two coupled algebraic equations for \n\nN\n=\n2\n\n, namely,
Now, from the relation \n\n\nx\n1\nL\n\n\nx\n2\nL\n\n=\n1\n\n we can eliminate one of the unknowns in (19)—for instance, by setting \n\n\nx\n2\n\n=\n\nω\na\n\n/\n\nx\n1\n\n\n, where \n\n\nω\na\n\n=\nexp\n\n\n2\nπia\n/\nL\n\n\n\n, \n\n1\n⩽\na\n⩽\nL\n\n, are the roots of unity of degree \n\nL\n\n. Replacing these values for \n\n\nx\n2\n\n\n into (19), we obtain the following polynomial equations fixing \n\n\nx\n1\n\n\n:
We can easily verify that the polynomial \n\n\np\na\n\n\nz\n\n\n is SI for each value of \n\na\n\n. They also satisfy the relations \n\n\np\na\n\n\nz\n\n=\n\nz\nL\n\np\n\n\n\nω\na\n\n/\nz\n\n\n\n, \n\n1\n⩽\na\n⩽\nL\n\n, which means that the solutions of (19) have the general form \n\nX\n=\n\nζ\n\n\nω\na\n\n/\nζ\n\n\n\n for \n\nζ\n\n any zero of \n\n\np\na\n\n\nz\n\n\n. In [76], the distribution of the zeros of the polynomials \n\n\np\na\n\n\nz\n\n\n was analyzed through an application of Vieira’s Theorem 25. It was shown that the exact behavior of the zeros of the polynomials \n\n\np\na\n\n\nz\n\n\n, for each \n\na\n\n, depends on two critical values of \n\nΔ\n\n, namely
as follows: if \n\n∣\nΔ\n∣\n⩽\n\nΔ\na\n\n1\n\n\n\n, then all the zeros of \n\n\np\na\n\n\nz\n\n\n are on \n\nS\n\n; if \n\n∣\nΔ\n∣\n⩾\n\nΔ\na\n\n2\n\n\n\n, then all the zeros of \n\n\np\na\n\n\nz\n\n\n but two are on \n\nS\n\n; (see [76] for the case \n\n\nΔ\na\n\n1\n\n\n<\n∣\nΔ\n∣\n<\n\nΔ\na\n\n2\n\n\n\n and more details).
\n
Finally, we highlight that the polynomial \n\n\np\na\n\n\nz\n\n\n becomes a Salem polynomial for \n\na\n=\nL\n\n and integer values of \n\nΔ\n\n. This was one of the first appearances of Salem polynomials in physics.
\n
\n
\n
5.4 Orthogonal polynomials
\n
An infinite sequence \n\nP\n=\n\n\n\n\nP\nn\n\n\nz\n\n\n\n\nn\n∈\nN\n\n\n\n of polynomials \n\n\nP\nn\n\n\nz\n\n\n of degree \n\nn\n\n is said to be an orthogonal polynomial sequence on the interval \n\n\nl\nr\n\n\n of the real line if there exists a function \n\nw\n\nx\n\n\n, positive in \n\n\nl\nr\n\n∈\nR\n\n, such that
where \n\n\nK\n0\n\n\n, \n\n\nK\n1\n\n\n, etc. are positive numbers. Orthogonal polynomial sequences on the real line have many interesting and important properties—see [77].
\n
Very recently, Vieira and Botta [78, 79] studied the action of Möbius transformations over orthogonal polynomial sequences on the real line. In particular, they showed that the infinite sequence \n\nT\n=\n\n\n\n\nT\nn\n\n\nz\n\n\n\n\nn\n∈\nN\n\n\n\n of the Möbius-transformed polynomials \n\n\nT\nn\n\n\nz\n\n=\n\n\n\nz\n−\n1\n\n\nn\n\n\nP\nn\n\n\n\nW\n\nz\n\n\n\n\n, where \n\nW\n\nz\n\n=\n−\ni\n\n\nz\n+\n1\n\n\n/\n\n\nz\n−\n1\n\n\n\n, is an SI polynomial sequence with all their zeros on the unit circle \n\nS\n\n—see Table 2 for an example. We highlight that the polynomials \n\n\nT\nn\n\n\nz\n\n∈\nT\n\n also have properties similar to the original polynomials \n\n\nP\nn\n\n\nz\n\n∈\nP\n\n as, for instance, they satisfy an orthogonality condition on the unit circle and a three-term recurrence relation, their zeros lie all on \n\nS\n\n and are simple, for \n\nn\n⩾\n1\n\n the zeros of \n\n\nT\nn\n\n\nz\n\n\n interlaces with those of \n\n\nT\n\nn\n+\n1\n\n\n\nz\n\n\n and so on—see [78, 79] for more details.
Hermite and Möbius-transformed Hermite polynomials, up to \n\n4\n\nth degree.
\n
\n
\n
\n
6. Conclusions
\n
In this work, we reviewed the theory of self-conjugate, self-reciprocal, and self-inversive polynomials. We discussed their main properties, how they are related to each other, the main theorems regarding the distribution of their zeros and some applications of these polynomials both in physics and mathematics. We hope that this short review suits for a compact introduction of the subject, paving the way for further developments in this interesting field of research.
\n
\n
Acknowledgments
\n
We thank the editorial staff for all the support during the publishing process and also the Coordination for the Improvement of Higher Education (CAPES).
\n
\n',keywords:"self-inversive polynomials, self-reciprocal polynomials, Pisot and Salem polynomials, Möbius transformations, knot theory, Bethe equations",chapterPDFUrl:"https://cdn.intechopen.com/pdfs/66332.pdf",chapterXML:"https://mts.intechopen.com/source/xml/66332.xml",downloadPdfUrl:"/chapter/pdf-download/66332",previewPdfUrl:"/chapter/pdf-preview/66332",totalDownloads:250,totalViews:163,totalCrossrefCites:0,dateSubmitted:"August 11th 2018",dateReviewed:"November 26th 2018",datePrePublished:"March 23rd 2019",datePublished:null,readingETA:"0",abstract:"Polynomials whose zeros are symmetric either to the real line or to the unit circle are very important in mathematics and physics. We can classify them into three main classes: the self-conjugate polynomials, whose zeros are symmetric to the real line; the self-inversive polynomials, whose zeros are symmetric to the unit circle; and the self-reciprocal polynomials, whose zeros are symmetric by an inversion with respect to the unit circle followed by a reflection in the real line. Real self-reciprocal polynomials are simultaneously self-conjugate and self-inversive so that their zeros are symmetric to both the real line and the unit circle. In this survey, we present a short review of these polynomials, focusing on the distribution of their zeros.",reviewType:"peer-reviewed",bibtexUrl:"/chapter/bibtex/66332",risUrl:"/chapter/ris/66332",signatures:"Ricardo Vieira",book:{id:"8599",title:"Polynomials",subtitle:"Theory and Application",fullTitle:"Polynomials - Theory and Application",slug:"polynomials-theory-and-application",publishedDate:"May 2nd 2019",bookSignature:"Cheon Seoung Ryoo",coverURL:"https://cdn.intechopen.com/books/images_new/8599.jpg",licenceType:"CC BY 3.0",editedByType:"Edited by",editors:[{id:"230100",title:"Prof.",name:"Cheon Seoung",middleName:null,surname:"Ryoo",slug:"cheon-seoung-ryoo",fullName:"Cheon Seoung Ryoo"}],productType:{id:"1",title:"Edited Volume",chapterContentType:"chapter",authoredCaption:"Edited by"}},authors:null,sections:[{id:"sec_1",title:"1. Introduction",level:"1"},{id:"sec_2",title:"2. Self-conjugate, self-reciprocal, and self-inversive polynomials",level:"1"},{id:"sec_3",title:"3. How these polynomials are related to each other?",level:"1"},{id:"sec_4",title:"4. Zeros location theorems",level:"1"},{id:"sec_4_2",title:"4.1 Polynomials that do not necessarily have symmetric zeros",level:"2"},{id:"sec_5_2",title:"4.2 Real self-reciprocal polynomials",level:"2"},{id:"sec_6_2",title:"4.3 Complex self-reciprocal and self-inversive polynomials",level:"2"},{id:"sec_8",title:"5. Where these polynomials are found?",level:"1"},{id:"sec_8_2",title:"5.1 Polynomials with small Mahler measure",level:"2"},{id:"sec_9_2",title:"5.2 Knot theory",level:"2"},{id:"sec_10_2",title:"5.3 Bethe equations",level:"2"},{id:"sec_11_2",title:"5.4 Orthogonal polynomials",level:"2"},{id:"sec_13",title:"6. Conclusions",level:"1"},{id:"sec_14",title:"Acknowledgments",level:"1"}],chapterReferences:[{id:"B1",body:'Marden M. Geometry of Polynomials. 2nd ed. Vol. 3. Providence, Rhode Island: American Mathematical Society; 1966\n'},{id:"B2",body:'Milovanović GV, Mitrinović DS, Rassias TM. Topics in Polynomials: Extremal Problems, Inequalities, Zeros. Singapore: World Scientific; 1994\n'},{id:"B3",body:'Sheil-Small T. Complex Polynomials. Vol. 75. Cambridge: Cambridge University Press; 2002\n'},{id:"B4",body:'Vieira R. How to count the number of zeros that a polynomial has on the unit circle? 2019. arXiv preprint: arXiv:1902.04231\n'},{id:"B5",body:'Cohn A. Über die Anzahl der Wurzeln einer algebraischen Gleichung in einem Kreise. Mathematische Zeitschrift. 1922;14(1):110-148. DOI: 10.1007/BF01215894\n'},{id:"B6",body:'Bonsall F, Marden M. Zeros of self-inversive polynomials. Proceedings of the American Mathematical Society. 1952;3(3):471-475. DOI: 10.2307/2031905\n'},{id:"B7",body:'Ancochea G. Zeros of self-inversive polynomials. Proceedings of the American Mathematical Society. 1953;4(6):900-902. DOI: 10.2307/2031826\n'},{id:"B8",body:'Eneström G. Härledning af en allmän formel för antalet pensionärer som vid en godtycklig tidpunkt förefinnas inom en sluten pensionskassa. Öfversigt af Kongl. Vetenskaps-Akademiens Förhandlingar. 1893;50:405-415\n'},{id:"B9",body:'Eneström G. Remarque sur un théorème relatif aux racines de l’équation \n\n\na\nn\n\n\nx\nn\n\n+\n\na\n\nn\n−\n1\n\n\n\nx\n\nn\n−\n1\n\n\n+\n⋯\n+\n\na\n1\n\nx\n+\n\na\n0\n\n=\n0\n\n où tous les coefficientes a sont réels et positifs. Tohoku Mathematical Journal, First Series. 1920;18:34-36\n'},{id:"B10",body:'Kakeya S. On the limits of the roots of an algebraic equation with positive coefficients. Tohoku Mathematical Journal, First Series. 1912;2:140-142\n'},{id:"B11",body:'Jury E. A note on the reciprocal zeros of a real polynomial with respect to the unit circle. IEEE Transactions on Circuit Theory. 1964;11(2):292-294. DOI: 10.1109/TCT.1964.1082289\n'},{id:"B12",body:'Chen W. On the polynomials with all their zeros on the unit circle. Journal of Mathematical Analysis and Applications. 1995;190(3):714-724. DOI: 10.1006/jmaa.1995.1105\n'},{id:"B13",body:'Krein M, Naimark M. The method of symmetric and Hermitian forms in the theory of the separation of the roots of algebraic equations. Linear and Multilinear Algebra. 1981;10(4):265-308. DOI: 10.1080/03081088108817420\n'},{id:"B14",body:'Chinen K. An abundance of invariant polynomials satisfying the Riemann hypothesis. Discrete Mathematics. 2008;308(24):6426-6440. DOI: 10.1016/j.disc.2007.12.022\n'},{id:"B15",body:'Choo Y. On the zeros of a family of self-reciprocal polynomials. International Journal of Mathematical Analysis. 2011;5(36):1761-1766\n'},{id:"B16",body:'Lakatos P. On zeros of reciprocal polynomials. Publicationes Mathematicae Debrecen. 2002;61(3–4):645-661\n'},{id:"B17",body:'Lakatos P, Losonczi L. On zeros of reciprocal polynomials of odd degree. Journal of Inequalities in Pure and Applied Mathematics. 2003;4(3):8-15\n'},{id:"B18",body:'Lakatos P, Losonczi L. Circular interlacing with reciprocal polynomials. Mathematical Inequalities & Applications. 2007;10(4):761. DOI: 10.7153/mia-10-71\n'},{id:"B19",body:'Kwon DY. Reciprocal polynomials with all zeros on the unit circle. Acta Mathematica Hungarica. 2011;131(3):285-294. DOI: 10.1007/s10474–011–0090–6\n'},{id:"B20",body:'Kwon DY. Reciprocal polynomials with all but two zeros on the unit circle. Acta Mathematica Hungarica. 2011;134(4):472-480. DOI: 10.1007/s10474–011–0176–1\n'},{id:"B21",body:'Konvalina J, Matache V. Palindrome-polynomials with roots on the unit circle. Comptes Rendus Mathematique. 2004;26(2):39\n'},{id:"B22",body:'Kim S-H, Park CW. On the zeros of certain self-reciprocal polynomials. Journal of Mathematical Analysis and Applications. 2008;339(1):240-247. DOI: 10.1016/j.jmaa.2007.06.055\n'},{id:"B23",body:'Kim S-H, Lee JH. On the zeros of self-reciprocal polynomials satisfying certain coefficient conditions. Bulletin of the Korean Mathematical Society. 2010;47(6):1189-1194. DOI: 10.4134/BKMS.2010.47.6.1189\n'},{id:"B24",body:'Botta V, Bracciali CF, Pereira JA. Some properties of classes of real self-reciprocal polynomials. Journal of Mathematical Analysis and Applications. 2016;433(2):1290-1304. DOI: 10.1016/j.jmaa.2015.08.038\n'},{id:"B25",body:'Suzuki M. On zeros of self-reciprocal polynomials. 2012. ArXiv: ArXiv:1211.2953\n'},{id:"B26",body:'Botta V, Marques LF, Meneguette M. Palindromic and perturbed polynomials: Zeros location. Acta Mathematica Hungarica. 2014;143(1):81-87. DOI: 10.1007/s10474–013–0382–0\n'},{id:"B27",body:'Conrey B, Granville A, Poonen B, Soundararajan K. Zeros of Fekete polynomials. Annales de l’institut Fourier. 2000;50(3):865-890. DOI: 10.5802/aif.1776\n'},{id:"B28",body:'Erdélyi T. On the zeros of polynomials with Littlewood-type coefficient constraints. The Michigan Mathematical Journal. 2001;49(1):97-111. DOI: 10.1307/mmj/1008719037\n'},{id:"B29",body:'Mossinghoff MJ. Polynomials with restricted coefficients and prescribed non-cyclotomic factors. LMS Journal of Computation and Mathematics. 2003;6:314-325. DOI: 10.1112/S1461157000000474\n'},{id:"B30",body:'Mercer ID. Unimodular roots of special Littlewood polynomials. Canadian Mathematical Bulletin. 2006;49(3):438-447. DOI: 10.4153/CMB-2006–043-x\n'},{id:"B31",body:'Mukunda K. Littlewood Pisot numbers. Journal of Number Theory. 2006;117(1):106-121. DOI: 10.1016/j.jnt.2005.05.009\n'},{id:"B32",body:'Drungilas P. Unimodular roots of reciprocal Littlewood polynomials. Journal of the Korean Mathematical Society. 2008;45(3):835-840. DOI: 10.4134/JKMS.2008.45.3.835\n'},{id:"B33",body:'Baradaran J, Taghavi M. Polynomials with coefficients from a finite set. Mathematica Slovaca. 2014;64(6):1397-1408. DOI: 10.2478/s12175–014–0282-y\n'},{id:"B34",body:'Borwein P, Choi S, Ferguson R, Jankauskas J. On Littlewood polynomials with prescribed number of zeros inside the unit disk. Canadian Journal of Mathematics. 2015;67(3):507-526. DOI: 10.4153/CJM-2014–007–1\n'},{id:"B35",body:'Drungilas P, Jankauskas J, Šiurys J. On Littlewood and Newman polynomial multiples of Borwein polynomials. Mathematics of Computation. 2018;87(311):1523-1541. DOI: 10.1090/mcom/3258\n'},{id:"B36",body:'Odlyzko A, Poonen B. Zeros of polynomials with 0,1 coefficients. L’Enseignement Mathématique. 1993;39:317-348\n'},{id:"B37",body:'Murty MR, Smyth C, Wang RJ. Zeros of Ramanujan polynomials. Journal of the Ramanujan Mathematical Society. 2011;26(1):107-125\n'},{id:"B38",body:'Lalín MN, Rogers MD, et al. Variations of the Ramanujan polynomials and remarks on \n\nζ\n\n\n2\nj\n+\n1\n\n\n/\n\nπ\n\n2\nj\n+\n1\n\n\n\n. Functiones et Approximatio Commentarii Mathematici. 2013;48(1):91-111. DOI: 10.7169/facm/2013.48.1.8\n'},{id:"B39",body:'Diamantis N, Rolen L. Period polynomials, derivatives of L-functions, and zeros of polynomials. Research in the Mathematical Sciences. 2018;5(1):9. DOI: 10.1007/s40687–018–0126–4\n'},{id:"B40",body:'Lindstrøm P. Galois Theory of Palindromic Polynomials. Oslo: University of Oslo; 2015\n'},{id:"B41",body:'O’Hara PJ, Rodriguez RS. Some properties of self-inversive polynomials. Proceedings of the American Mathematical Society. 1974;44(2):331-335. DOI: 10.1090/S0002–9939–1974–0349967–5\n'},{id:"B42",body:'Schinzel A. Self-inversive polynomials with all zeros on the unit circle. The Ramanujan Journal. 2005;9(1):19-23. DOI: 10.1007/s11139–005–0821–9\n'},{id:"B43",body:'Losonczi L, Schinzel A. Self-inversive polynomials of odd degree. The Ramanujan Journal. 2007;14(2):305-320. DOI: 10.1007/s11139–007–9029–5\n'},{id:"B44",body:'Lakatos P, Losonczi L. Self-inversive polynomials whose zeros are on the unit circle. Publicationes Mathematicae Debrecen. 2004;65(3–4):409-420\n'},{id:"B45",body:'Lakatos P, Losonczi L. Polynomials with all zeros on the unit circle. Acta Mathematica Hungarica. 2009;125(4):341-356. DOI: 10.1007/s10474–009–9028–7\n'},{id:"B46",body:'Losonczi L. On reciprocal polynomials with zeros of modulus one. Mathematical Inequalities & Applications. 2006;9(2):289. DOI: 10.7153/mia-09–29\n'},{id:"B47",body:'Vieira RS. On the number of roots of self-inversive polynomials on the complex unit circle. The Ramanujan Journal. 2017;42(2):363-369. DOI: 10.1007/s11139–016–9804–2\n'},{id:"B48",body:'Stankov D. The necessary and sufficient condition for an algebraic integer to be a Salem number. 2018. arXiv:1706.01767\n'},{id:"B49",body:'Sinclair C, Vaaler J. Self-inversive polynomials with all zeros on the unit circle. In: McKee J, Smyth C, editors. Number Theory and Polynomials. London Mathematical Society Lecture Note Series. Cambridge: Cambridge University Press; 2008. pp. 312-321. DOI: 10.1017/CB09780511721274.020\n'},{id:"B50",body:'Choo Y, Kim Y-J. On the zeros of self-inversive polynomials. International Journal of Mathematical Analysis. 2013;7:187-193\n'},{id:"B51",body:'Area I, Godoy E, Lamblém RL, Ranga AS. Basic hypergeometric polynomials with zeros on the unit circle. Applied Mathematics and Computation. 2013;225:622-630. DOI: 10.1016/j.amc.2013.09.060\n'},{id:"B52",body:'Dimitrov D, Ismail M, Ranga AS. A class of hypergeometric polynomials with zeros on the unit circle: Extremal and orthogonal properties and quadrature formulas. Applied Numerical Mathematics. 2013;65:41-52\n'},{id:"B53",body:'Kim E. A family of self-inversive polynomials with concyclic zeros. Journal of Mathematical Analysis and Applications. 2013;401(2):695-701. DOI: 10.1016/j.jmaa.2012.12.048\n'},{id:"B54",body:'Ito N, Wimmer HK. Self-inversive Hilbert space operator polynomials with spectrum on the unit circle. Journal of Mathematical Analysis and Applications. 2016;436(2):683-691. DOI: 10.4153/CMB-2001–044-x\n'},{id:"B55",body:'Joyner D, Kim J-L. Selected Unsolved Problems in Coding Theory. Birkhäuser Basel, New York: Springer Science & Business Media; 2011. DOI: 10.1007/978–0-8176–8256–9\n'},{id:"B56",body:'Joyner D. Zeros of some self-reciprocal polynomials. In: Excursions in Harmonic Analysis. Vol. 1. Birkhäuser, Boston: Springer; 2013. pp. 329-348. DOI: 10.1007/978–0-8176–8376–4_17\n'},{id:"B57",body:'Joyner D, Shaska T. Self-inversive polynomials, curves, and codes. In: Higher Genus Curves in Mathematical Physics and Arithmetic Geometry. Vol. 703. American Mathematical Society; 2018. pp. 189-208. DOI: 10.1090/conm/703\n'},{id:"B58",body:'McKee J, McKee JF, Smyth C. Number Theory and Polynomials. Vol. 352. Cambridge: Cambridge University Press; 2008\n'},{id:"B59",body:'Lee T-D, Yang C-N. Statistical theory of equations of state and phase transitions II. Lattice gas and Ising model. Physical Review. 1952;87(3):410. DOI: 10.1103/PhysRev.87.410\n'},{id:"B60",body:'He Y-H. Polynomial roots and Calabi-Yau geometries. Advances in High Energy Physics. 2011;2011:1-15. DOI: 10.1155/2011/719672\n'},{id:"B61",body:'Everest G, Ward T. Heights of Polynomials and Entropy in Algebraic Dynamics. Springer-Verlag, London: Springer Science & Business Media; 2013\n'},{id:"B62",body:'Bertin MJ, Decomps-Guilloux A, Grandet-Hugot M, Pathiaux-Delefosse M, Schreiber J. Pisot and Salem Numbers. Birkhäuser, Basel: Birkhäuser; 2012\n'},{id:"B63",body:'Smyth CJ. On the product of the conjugates outside the unit circle of an algebraic integer. Bulletin of the London Mathematical Society. 1971;3(2):169-175. DOI: 10.1112/blms/3.2.169\n'},{id:"B64",body:'Smyth C. Seventy years of Salem numbers. Bulletin of the London Mathematical Society. 2015;47(3):379-395. DOI: 10.1112/blms/bdv027\n'},{id:"B65",body:'Boyd DW. Small Salem numbers. Duke Mathematical Journal. 1977;44(2):315-328. DOI: 10.1215/S0012–7094–77–04413–1\n'},{id:"B66",body:'Mossinghoff M. Polynomials with small Mahler measure. Mathematics of Computation of the American Mathematical Society. 1998;67(224):1697-1705. DOI: 10.1090/S0025–5718–98–01006–0\n'},{id:"B67",body:'Lehmer DH. Factorization of certain cyclotomic functions. Annals of Mathematics. 1933;34(3):461-479. DOI: 10.2307/1968172\n'},{id:"B68",body:'Hironaka E. What is… Lehmer’s number? Notices of the American Mathematical Society; 2009;56:374-375\n'},{id:"B69",body:'Bailey DH, Broadhurst DJ. A seventeenth-order polylogarithm ladder. 1999. arXiv preprint: math/9906134\n'},{id:"B70",body:'Adams CC. The Knot Book: An Elementary Introduction to the Mathematical Theory of Knots. Providence: American Mathematical Society; 2004\n'},{id:"B71",body:'Alexander JW. Topological invariants of knots and links. Transactions of the American Mathematical Society. 1928;30(2):275-306. DOI: 10.1090/S0002–9947–1928–1501429–1\n'},{id:"B72",body:'Jones V. A polynomial invariant for knots via von Neumann algebras. Bulletin of the American Mathematical Society. 1985;12:103-111\n'},{id:"B73",body:'Hironaka E. The Lehmer polynomial and pretzel links. Canadian Mathematical Bulletin. 2001;44(4):440-451\n'},{id:"B74",body:'Vieira R. Solving and classifying the solutions of the Yang-Baxter equation through a differential approach. Two-state systems. Journal of High Energy Physics. 2018;2018(10):110. DOI: 10.1007/JHEP10(2018)110\n'},{id:"B75",body:'Bethe H. Zur theorie der metalle. Zeitschrift für Physik. 1931;71(3–4):205-226\n'},{id:"B76",body:'Vieira RS, Lima-Santos A. Where are the roots of the Bethe Ansatz equations? Physics Letters A. 2015;379(37):2150-2153. DOI: 10.1016/j.physleta.2015.07.016\n'},{id:"B77",body:'Chihara TS. An Introduction to Orthogonal Polynomials. Dover Publications; 2011\n'},{id:"B78",body:'Vieira RS, Botta V. Möbius transformations and orthogonal polynomials (In preparation). New York: Gordon and Breach Science Publishers; 1978\n'},{id:"B79",body:'Vieira RS, Botta V. Möbius transformations, orthogonal polynomials and self-inversive polynomials (In preparation). New York: Gordon and Breach Science Publishers; 1978\n'}],footnotes:[{id:"fn1",explanation:"The reader should be aware that there is no standard in naming these polynomials. For instance, what we call here self-inversive polynomials are sometimes called self-reciprocal polynomials. What we mean positive self-reciprocal polynomials are usually just called self-reciprocal or yet palindrome polynomials (because their coefficients are the same whether they are read from forwards or backwards), as well as, negative self-reciprocal polynomials are usually called skew-reciprocal, anti-reciprocal, or yet anti-palindrome polynomials."},{id:"fn2",explanation:"The zeros of such polynomials present a fractal behavior, as was first discovered by Odlyzko and Poonen in [36]."},{id:"fn3",explanation:"The Mahler measure of a monic integer polynomial \n\np\n\nz\n\n\n can also be defined without making reference to its zeros through the formula \n\nM\n\n\np\n\nz\n\n\n\n=\nexp\n\n\n\n∫\n0\n1\n\nlog\n\n\np\n\n\ne\n\n2\nπit\n\n\n\n\n\ndt\n\n\n\n—see [61]."},{id:"fn4",explanation:"We remark, however, that different knots can have the same knot invariant. Up to date, we do not know whether there exists a knot invariant that distinguishes all non-equivalent knots from each other (although there do exist some invariants that distinguish every knot from the trivial knot). Thus, until now the concept of knot invariants only partially solves the problem of knot classification."},{id:"fn5",explanation:"Alexander polynomials can also be defined as Laurent polynomials, see [70]."}],contributors:[{corresp:"yes",contributorFullName:"Ricardo Vieira",address:"rs.vieira@unesp.br",affiliation:'
Faculty of Science and Technology, Department of Mathematics and Computer Science, São Paulo State University (UNESP), Presidente Prudente, SP, Brazil
'}],corrections:null},book:{id:"8599",title:"Polynomials",subtitle:"Theory and Application",fullTitle:"Polynomials - Theory and Application",slug:"polynomials-theory-and-application",publishedDate:"May 2nd 2019",bookSignature:"Cheon Seoung Ryoo",coverURL:"https://cdn.intechopen.com/books/images_new/8599.jpg",licenceType:"CC BY 3.0",editedByType:"Edited by",editors:[{id:"230100",title:"Prof.",name:"Cheon Seoung",middleName:null,surname:"Ryoo",slug:"cheon-seoung-ryoo",fullName:"Cheon Seoung Ryoo"}],productType:{id:"1",title:"Edited Volume",chapterContentType:"chapter",authoredCaption:"Edited by"}}},profile:{item:{id:"66011",title:"Prof.",name:"Masaru",middleName:null,surname:"Ishizuka",email:"ishizuka@pu-toyama.ac.jp",fullName:"Masaru Ishizuka",slug:"masaru-ishizuka",position:null,biography:null,institutionString:null,profilePictureURL:"//cdnintech.com/web/frontend/www/assets/author.svg",totalCites:0,totalChapterViews:"0",outsideEditionCount:0,totalAuthoredChapters:"1",totalEditedBooks:"0",personalWebsiteURL:null,twitterURL:null,linkedinURL:null,institution:{name:"Toyama Prefectural University",institutionURL:null,country:{name:"Japan"}}},booksEdited:[],chaptersAuthored:[{title:"Design of Electronic Equipment Casings for Natural Air Cooling: Effects of Height and Size of Outlet Vent on Flow Resistance",slug:"design-of-electronic-equipment-casings-for-natural-air-cooling-effects-of-height-and-size-of-outlet-",abstract:null,signatures:"Masaru Ishizuka and Tomoyuki Hatakeyama",authors:[{id:"66011",title:"Prof.",name:"Masaru",surname:"Ishizuka",fullName:"Masaru Ishizuka",slug:"masaru-ishizuka",email:"ishizuka@pu-toyama.ac.jp"},{id:"66518",title:"Dr.",name:"Tomoyuki",surname:"Hatakeyama",fullName:"Tomoyuki Hatakeyama",slug:"tomoyuki-hatakeyama",email:"hatake@pu-toyama.ac.jp"}],book:{title:"Heat Transfer",slug:"heat-transfer-engineering-applications",productType:{id:"1",title:"Edited Volume"}}}],collaborators:[{id:"65241",title:"Prof.",name:"Shigeru",surname:"Nozu",slug:"shigeru-nozu",fullName:"Shigeru Nozu",position:null,profilePictureURL:"//cdnintech.com/web/frontend/www/assets/author.svg",biography:null,institutionString:null,institution:{name:"Okayama Prefectural University",institutionURL:null,country:{name:"Japan"}}},{id:"66973",title:"Dr.",name:"Michał",surname:"Szymański",slug:"michal-szymanski",fullName:"Michał Szymański",position:null,profilePictureURL:"//cdnintech.com/web/frontend/www/assets/author.svg",biography:null,institutionString:null,institution:{name:"Institute of Electron Technology",institutionURL:null,country:{name:"Poland"}}},{id:"67663",title:"Prof.",name:"Alejandro",surname:"Crespo-Sosa",slug:"alejandro-crespo-sosa",fullName:"Alejandro Crespo-Sosa",position:null,profilePictureURL:"//cdnintech.com/web/frontend/www/assets/author.svg",biography:null,institutionString:null,institution:{name:"National Autonomous University of Mexico",institutionURL:null,country:{name:"Mexico"}}},{id:"68268",title:"Prof.",name:"Nicolas",surname:"Kazansky",slug:"nicolas-kazansky",fullName:"Nicolas Kazansky",position:null,profilePictureURL:"//cdnintech.com/web/frontend/www/assets/author.svg",biography:null,institutionString:null,institution:{name:"Image Processing Systems Institute",institutionURL:null,country:{name:"Russia"}}},{id:"71737",title:"Dr.",name:"Vsevolod",surname:"Kolpakov",slug:"vsevolod-kolpakov",fullName:"Vsevolod Kolpakov",position:null,profilePictureURL:"https://mts.intechopen.com/storage/users/71737/images/204_n.jpg",biography:null,institutionString:null,institution:{name:"Samara State Aerospace University",institutionURL:null,country:{name:"Russia"}}},{id:"72547",title:"Dr.",name:"Etsuji",surname:"Ohmura",slug:"etsuji-ohmura",fullName:"Etsuji Ohmura",position:null,profilePictureURL:"//cdnintech.com/web/frontend/www/assets/author.svg",biography:null,institutionString:null,institution:{name:"Osaka University",institutionURL:null,country:{name:"Japan"}}},{id:"72957",title:"Dr.",name:"Gasser",surname:"Abdelal",slug:"gasser-abdelal",fullName:"Gasser Abdelal",position:null,profilePictureURL:"//cdnintech.com/web/frontend/www/assets/author.svg",biography:null,institutionString:null,institution:{name:"University of Liverpool",institutionURL:null,country:{name:"United Kingdom"}}},{id:"73688",title:"Prof.",name:"Hiroaki",surname:"Tsuji",slug:"hiroaki-tsuji",fullName:"Hiroaki Tsuji",position:null,profilePictureURL:"//cdnintech.com/web/frontend/www/assets/author.svg",biography:null,institutionString:null,institution:{name:"Okayama Prefectural University",institutionURL:null,country:{name:"Japan"}}},{id:"73800",title:"Dr.",name:"Kenji",surname:"Onishi",slug:"kenji-onishi",fullName:"Kenji Onishi",position:null,profilePictureURL:"//cdnintech.com/web/frontend/www/assets/author.svg",biography:null,institutionString:null,institution:null},{id:"75174",title:"Dr.",name:"David",surname:"Sands",slug:"david-sands",fullName:"David Sands",position:null,profilePictureURL:"//cdnintech.com/web/frontend/www/assets/author.svg",biography:null,institutionString:null,institution:{name:"University of Hull",institutionURL:null,country:{name:"United Kingdom"}}}]},generic:{page:{slug:"contact-us",title:"Contact Us",intro:"
We pride ourselves on our belief that scientific progress is generated by collaboration, that the playing field for scientific research should be leveled globally, and that research conducted in a democratic environment, with the use of innovative technologies, should be made available to anyone.
\n\n
We look forward to hearing from individuals and organizations who are interested in new discoveries and sharing their research.
*INTECHOPEN LIMITED is a privately owned company registered in England and Wales, No. 11086078 Registered Office: 7th floor, 10 Lower Thames Street, London, EC3R 6AF, UK
*INTECHOPEN LIMITED is a privately owned company registered in England and Wales, No. 11086078 Registered Office: 7th floor, 10 Lower Thames Street, London, EC3R 6AF, UK
\n'}]},successStories:{items:[]},authorsAndEditors:{filterParams:{sort:"featured,name"},profiles:[{id:"6700",title:"Dr.",name:"Abbass A.",middleName:null,surname:"Hashim",slug:"abbass-a.-hashim",fullName:"Abbass A. Hashim",position:null,profilePictureURL:"https://mts.intechopen.com/storage/users/6700/images/1864_n.jpg",biography:"Currently I am carrying out research in several areas of interest, mainly covering work on chemical and bio-sensors, semiconductor thin film device fabrication and characterisation.\nAt the moment I have very strong interest in radiation environmental pollution and bacteriology treatment. The teams of researchers are working very hard to bring novel results in this field. I am also a member of the team in charge for the supervision of Ph.D. students in the fields of development of silicon based planar waveguide sensor devices, study of inelastic electron tunnelling in planar tunnelling nanostructures for sensing applications and development of organotellurium(IV) compounds for semiconductor applications. I am a specialist in data analysis techniques and nanosurface structure. I have served as the editor for many books, been a member of the editorial board in science journals, have published many papers and hold many patents.",institutionString:null,institution:{name:"Sheffield Hallam University",country:{name:"United Kingdom"}}},{id:"54525",title:"Prof.",name:"Abdul Latif",middleName:null,surname:"Ahmad",slug:"abdul-latif-ahmad",fullName:"Abdul Latif Ahmad",position:null,profilePictureURL:"//cdnintech.com/web/frontend/www/assets/author.svg",biography:null,institutionString:null,institution:null},{id:"20567",title:"Prof.",name:"Ado",middleName:null,surname:"Jorio",slug:"ado-jorio",fullName:"Ado Jorio",position:null,profilePictureURL:"//cdnintech.com/web/frontend/www/assets/author.svg",biography:null,institutionString:null,institution:{name:"Universidade Federal de Minas Gerais",country:{name:"Brazil"}}},{id:"47940",title:"Dr.",name:"Alberto",middleName:null,surname:"Mantovani",slug:"alberto-mantovani",fullName:"Alberto Mantovani",position:null,profilePictureURL:"//cdnintech.com/web/frontend/www/assets/author.svg",biography:null,institutionString:null,institution:null},{id:"12392",title:"Mr.",name:"Alex",middleName:null,surname:"Lazinica",slug:"alex-lazinica",fullName:"Alex Lazinica",position:null,profilePictureURL:"https://mts.intechopen.com/storage/users/12392/images/7282_n.png",biography:"Alex Lazinica is the founder and CEO of IntechOpen. After obtaining a Master's degree in Mechanical Engineering, he continued his PhD studies in Robotics at the Vienna University of Technology. Here he worked as a robotic researcher with the university's Intelligent Manufacturing Systems Group as well as a guest researcher at various European universities, including the Swiss Federal Institute of Technology Lausanne (EPFL). During this time he published more than 20 scientific papers, gave presentations, served as a reviewer for major robotic journals and conferences and most importantly he co-founded and built the International Journal of Advanced Robotic Systems- world's first Open Access journal in the field of robotics. Starting this journal was a pivotal point in his career, since it was a pathway to founding IntechOpen - Open Access publisher focused on addressing academic researchers needs. Alex is a personification of IntechOpen key values being trusted, open and entrepreneurial. Today his focus is on defining the growth and development strategy for the company.",institutionString:null,institution:{name:"TU Wien",country:{name:"Austria"}}},{id:"19816",title:"Prof.",name:"Alexander",middleName:null,surname:"Kokorin",slug:"alexander-kokorin",fullName:"Alexander Kokorin",position:null,profilePictureURL:"https://mts.intechopen.com/storage/users/19816/images/1607_n.jpg",biography:"Alexander I. Kokorin: born: 1947, Moscow; DSc., PhD; Principal Research Fellow (Research Professor) of Department of Kinetics and Catalysis, N. Semenov Institute of Chemical Physics, Russian Academy of Sciences, Moscow.\nArea of research interests: physical chemistry of complex-organized molecular and nanosized systems, including polymer-metal complexes; the surface of doped oxide semiconductors. He is an expert in structural, absorptive, catalytic and photocatalytic properties, in structural organization and dynamic features of ionic liquids, in magnetic interactions between paramagnetic centers. The author or co-author of 3 books, over 200 articles and reviews in scientific journals and books. He is an actual member of the International EPR/ESR Society, European Society on Quantum Solar Energy Conversion, Moscow House of Scientists, of the Board of Moscow Physical Society.",institutionString:null,institution:null},{id:"62389",title:"PhD.",name:"Ali Demir",middleName:null,surname:"Sezer",slug:"ali-demir-sezer",fullName:"Ali Demir Sezer",position:null,profilePictureURL:"https://mts.intechopen.com/storage/users/62389/images/3413_n.jpg",biography:"Dr. Ali Demir Sezer has a Ph.D. from Pharmaceutical Biotechnology at the Faculty of Pharmacy, University of Marmara (Turkey). He is the member of many Pharmaceutical Associations and acts as a reviewer of scientific journals and European projects under different research areas such as: drug delivery systems, nanotechnology and pharmaceutical biotechnology. Dr. Sezer is the author of many scientific publications in peer-reviewed journals and poster communications. Focus of his research activity is drug delivery, physico-chemical characterization and biological evaluation of biopolymers micro and nanoparticles as modified drug delivery system, and colloidal drug carriers (liposomes, nanoparticles etc.).",institutionString:null,institution:{name:"Marmara University",country:{name:"Turkey"}}},{id:"61051",title:"Prof.",name:"Andrea",middleName:null,surname:"Natale",slug:"andrea-natale",fullName:"Andrea Natale",position:null,profilePictureURL:"//cdnintech.com/web/frontend/www/assets/author.svg",biography:null,institutionString:null,institution:null},{id:"100762",title:"Prof.",name:"Andrea",middleName:null,surname:"Natale",slug:"andrea-natale",fullName:"Andrea Natale",position:null,profilePictureURL:"//cdnintech.com/web/frontend/www/assets/author.svg",biography:null,institutionString:null,institution:{name:"St David's Medical Center",country:{name:"United States of America"}}},{id:"107416",title:"Dr.",name:"Andrea",middleName:null,surname:"Natale",slug:"andrea-natale",fullName:"Andrea Natale",position:null,profilePictureURL:"//cdnintech.com/web/frontend/www/assets/author.svg",biography:null,institutionString:null,institution:{name:"Texas Cardiac Arrhythmia",country:{name:"United States of America"}}},{id:"64434",title:"Dr.",name:"Angkoon",middleName:null,surname:"Phinyomark",slug:"angkoon-phinyomark",fullName:"Angkoon Phinyomark",position:null,profilePictureURL:"https://mts.intechopen.com/storage/users/64434/images/2619_n.jpg",biography:"My name is Angkoon Phinyomark. I received a B.Eng. degree in Computer Engineering with First Class Honors in 2008 from Prince of Songkla University, Songkhla, Thailand, where I received a Ph.D. degree in Electrical Engineering. My research interests are primarily in the area of biomedical signal processing and classification notably EMG (electromyography signal), EOG (electrooculography signal), and EEG (electroencephalography signal), image analysis notably breast cancer analysis and optical coherence tomography, and rehabilitation engineering. I became a student member of IEEE in 2008. During October 2011-March 2012, I had worked at School of Computer Science and Electronic Engineering, University of Essex, Colchester, Essex, United Kingdom. In addition, during a B.Eng. I had been a visiting research student at Faculty of Computer Science, University of Murcia, Murcia, Spain for three months.\n\nI have published over 40 papers during 5 years in refereed journals, books, and conference proceedings in the areas of electro-physiological signals processing and classification, notably EMG and EOG signals, fractal analysis, wavelet analysis, texture analysis, feature extraction and machine learning algorithms, and assistive and rehabilitative devices. I have several computer programming language certificates, i.e. Sun Certified Programmer for the Java 2 Platform 1.4 (SCJP), Microsoft Certified Professional Developer, Web Developer (MCPD), Microsoft Certified Technology Specialist, .NET Framework 2.0 Web (MCTS). I am a Reviewer for several refereed journals and international conferences, such as IEEE Transactions on Biomedical Engineering, IEEE Transactions on Industrial Electronics, Optic Letters, Measurement Science Review, and also a member of the International Advisory Committee for 2012 IEEE Business Engineering and Industrial Applications and 2012 IEEE Symposium on Business, Engineering and Industrial Applications.",institutionString:null,institution:{name:"Joseph Fourier University",country:{name:"France"}}},{id:"55578",title:"Dr.",name:"Antonio",middleName:null,surname:"Jurado-Navas",slug:"antonio-jurado-navas",fullName:"Antonio Jurado-Navas",position:null,profilePictureURL:"https://mts.intechopen.com/storage/users/55578/images/4574_n.png",biography:"Antonio Jurado-Navas received the M.S. degree (2002) and the Ph.D. degree (2009) in Telecommunication Engineering, both from the University of Málaga (Spain). He first worked as a consultant at Vodafone-Spain. From 2004 to 2011, he was a Research Assistant with the Communications Engineering Department at the University of Málaga. In 2011, he became an Assistant Professor in the same department. From 2012 to 2015, he was with Ericsson Spain, where he was working on geo-location\ntools for third generation mobile networks. Since 2015, he is a Marie-Curie fellow at the Denmark Technical University. His current research interests include the areas of mobile communication systems and channel modeling in addition to atmospheric optical communications, adaptive optics and statistics",institutionString:null,institution:{name:"University of Malaga",country:{name:"Spain"}}}],filtersByRegion:[{group:"region",caption:"North America",value:1,count:5313},{group:"region",caption:"Middle and South America",value:2,count:4819},{group:"region",caption:"Africa",value:3,count:1468},{group:"region",caption:"Asia",value:4,count:9362},{group:"region",caption:"Australia and Oceania",value:5,count:837},{group:"region",caption:"Europe",value:6,count:14778}],offset:12,limit:12,total:108153},chapterEmbeded:{data:{}},editorApplication:{success:null,errors:{}},ofsBooks:{filterParams:{sort:"dateEndThirdStepPublish",topicId:"23"},books:[{type:"book",id:"8452",title:"Argumentation",subtitle:null,isOpenForSubmission:!0,hash:"0860d5da35173d066fe692466ccc4487",slug:null,bookSignature:"",coverURL:"https://cdn.intechopen.com/books/images_new/8452.jpg",editedByType:null,editors:null,productType:{id:"1",chapterContentType:"chapter",authoredCaption:"Edited by"}},{type:"book",id:"8525",title:"LGBT+ Studies",subtitle:null,isOpenForSubmission:!0,hash:"6cb260b7524f548de18dac8fd97d5db7",slug:null,bookSignature:"",coverURL:"https://cdn.intechopen.com/books/images_new/8525.jpg",editedByType:null,editors:null,productType:{id:"1",chapterContentType:"chapter",authoredCaption:"Edited by"}},{type:"book",id:"7821",title:"Penology",subtitle:null,isOpenForSubmission:!0,hash:"2127505059817f762d95d38e83fa4b58",slug:null,bookSignature:"",coverURL:"https://cdn.intechopen.com/books/images_new/7821.jpg",editedByType:null,editors:null,productType:{id:"1",chapterContentType:"chapter",authoredCaption:"Edited by"}},{type:"book",id:"9036",title:"Social Policy",subtitle:null,isOpenForSubmission:!0,hash:"4266fa2d0a91696ade5421a773b40ec7",slug:null,bookSignature:"",coverURL:"https://cdn.intechopen.com/books/images_new/9036.jpg",editedByType:null,editors:null,productType:{id:"1",chapterContentType:"chapter",authoredCaption:"Edited by"}},{type:"book",id:"9037",title:"Types of Nonverbal Communication",subtitle:null,isOpenForSubmission:!0,hash:"9cc8207a08817f9049b2706066fb9ecd",slug:null,bookSignature:"",coverURL:"https://cdn.intechopen.com/books/images_new/9037.jpg",editedByType:null,editors:null,productType:{id:"1",chapterContentType:"chapter",authoredCaption:"Edited by"}},{type:"book",id:"9038",title:"Globalization",subtitle:null,isOpenForSubmission:!0,hash:"3761e38ada032e79742e548afe09a389",slug:null,bookSignature:"",coverURL:"https://cdn.intechopen.com/books/images_new/9038.jpg",editedByType:null,editors:null,productType:{id:"1",chapterContentType:"chapter",authoredCaption:"Edited by"}},{type:"book",id:"9041",title:"Active Reinforcement Learning",subtitle:null,isOpenForSubmission:!0,hash:"10d7900003c7af88a8d62946cd0dd920",slug:null,bookSignature:"",coverURL:"https://cdn.intechopen.com/books/images_new/9041.jpg",editedByType:null,editors:null,productType:{id:"1",chapterContentType:"chapter",authoredCaption:"Edited by"}},{type:"book",id:"9542",title:"Social Welfare",subtitle:null,isOpenForSubmission:!0,hash:"f87eca1e1da98d96aa76e55ab97eeb90",slug:null,bookSignature:"",coverURL:"https://cdn.intechopen.com/books/images_new/9542.jpg",editedByType:null,editors:null,productType:{id:"1",chapterContentType:"chapter",authoredCaption:"Edited by"}},{type:"book",id:"9538",title:"Generational Gap",subtitle:null,isOpenForSubmission:!0,hash:"9cf567fd09f3be48bf8dc6337fa13df6",slug:null,bookSignature:"",coverURL:"https://cdn.intechopen.com/books/images_new/9538.jpg",editedByType:null,editors:null,productType:{id:"1",chapterContentType:"chapter",authoredCaption:"Edited by"}},{type:"book",id:"9533",title:"Capitalism",subtitle:null,isOpenForSubmission:!0,hash:"5c20736ce60a5d627594d8036b8d64a7",slug:null,bookSignature:"",coverURL:"https://cdn.intechopen.com/books/images_new/9533.jpg",editedByType:null,editors:null,productType:{id:"1",chapterContentType:"chapter",authoredCaption:"Edited by"}},{type:"book",id:"9543",title:"Poverty",subtitle:null,isOpenForSubmission:!0,hash:"f1ced86a0f516c65e633cb59f944ae0e",slug:null,bookSignature:"",coverURL:"https://cdn.intechopen.com/books/images_new/9543.jpg",editedByType:null,editors:null,productType:{id:"1",chapterContentType:"chapter",authoredCaption:"Edited by"}},{type:"book",id:"9539",title:"Social Enterprise",subtitle:null,isOpenForSubmission:!0,hash:"ace77329fc7d282621439e01547ec5c0",slug:null,bookSignature:"",coverURL:"https://cdn.intechopen.com/books/images_new/9539.jpg",editedByType:null,editors:null,productType:{id:"1",chapterContentType:"chapter",authoredCaption:"Edited by"}}],filtersByTopic:[{group:"topic",caption:"Agricultural and Biological Sciences",value:5,count:35},{group:"topic",caption:"Biochemistry, Genetics and Molecular Biology",value:6,count:32},{group:"topic",caption:"Business, Management and Economics",value:7,count:9},{group:"topic",caption:"Chemistry",value:8,count:29},{group:"topic",caption:"Computer and Information Science",value:9,count:26},{group:"topic",caption:"Earth and Planetary Sciences",value:10,count:14},{group:"topic",caption:"Engineering",value:11,count:75},{group:"topic",caption:"Environmental Sciences",value:12,count:13},{group:"topic",caption:"Immunology and Microbiology",value:13,count:3},{group:"topic",caption:"Materials Science",value:14,count:37},{group:"topic",caption:"Mathematics",value:15,count:14},{group:"topic",caption:"Medicine",value:16,count:142},{group:"topic",caption:"Nanotechnology and Nanomaterials",value:17,count:5},{group:"topic",caption:"Neuroscience",value:18,count:6},{group:"topic",caption:"Pharmacology, Toxicology and Pharmaceutical Science",value:19,count:8},{group:"topic",caption:"Physics",value:20,count:20},{group:"topic",caption:"Psychology",value:21,count:2},{group:"topic",caption:"Robotics",value:22,count:6},{group:"topic",caption:"Social Sciences",value:23,count:14},{group:"topic",caption:"Technology",value:24,count:10},{group:"topic",caption:"Veterinary Medicine and Science",value:25,count:3},{group:"topic",caption:"Intelligent System",value:535,count:1}],offset:12,limit:12,total:28},popularBooks:{featuredBooks:[{type:"book",id:"7878",title:"Advances in Extracorporeal Membrane Oxygenation",subtitle:"Volume 3",isOpenForSubmission:!1,hash:"f95bf990273d08098a00f9a1c2403cbe",slug:"advances-in-extracorporeal-membrane-oxygenation-volume-3",bookSignature:"Michael S. Firstenberg",coverURL:"https://cdn.intechopen.com/books/images_new/7878.jpg",editors:[{id:"64343",title:null,name:"Michael S.",middleName:"S",surname:"Firstenberg",slug:"michael-s.-firstenberg",fullName:"Michael S. Firstenberg"}],productType:{id:"1",chapterContentType:"chapter"}},{type:"book",id:"7614",title:"Fourier Transforms",subtitle:"Century of Digitalization and Increasing Expectations",isOpenForSubmission:!1,hash:"ff3501657ae983a3b42fef1f7058ac91",slug:"fourier-transforms-century-of-digitalization-and-increasing-expectations",bookSignature:"Goran S. Nikoli? and Dragana Z. Markovi?-Nikoli?",coverURL:"https://cdn.intechopen.com/books/images_new/7614.jpg",editors:[{id:"23261",title:"Prof.",name:"Goran",middleName:"S.",surname:"Nikolic",slug:"goran-nikolic",fullName:"Goran Nikolic"}],productType:{id:"1",chapterContentType:"chapter"}},{type:"book",id:"8299",title:"Timber Buildings and Sustainability",subtitle:null,isOpenForSubmission:!1,hash:"bccf2891cec38ed041724131aa34c25a",slug:"timber-buildings-and-sustainability",bookSignature:"Giovanna Concu",coverURL:"https://cdn.intechopen.com/books/images_new/8299.jpg",editors:[{id:"108709",title:"Dr.",name:"Giovanna",middleName:null,surname:"Concu",slug:"giovanna-concu",fullName:"Giovanna Concu"}],productType:{id:"1",chapterContentType:"chapter"}},{type:"book",id:"7062",title:"Rhinosinusitis",subtitle:null,isOpenForSubmission:!1,hash:"14ed95e155b1e57a61827ca30b579d09",slug:"rhinosinusitis",bookSignature:"Balwant Singh Gendeh and Mirjana Turkalj",coverURL:"https://cdn.intechopen.com/books/images_new/7062.jpg",editors:[{id:"67669",title:"Prof.",name:"Balwant Singh",middleName:null,surname:"Gendeh",slug:"balwant-singh-gendeh",fullName:"Balwant Singh Gendeh"}],productType:{id:"1",chapterContentType:"chapter"}},{type:"book",id:"7087",title:"Tendons",subtitle:null,isOpenForSubmission:!1,hash:"786abac0445c102d1399a1e727a2db7f",slug:"tendons",bookSignature:"Hasan Sözen",coverURL:"https://cdn.intechopen.com/books/images_new/7087.jpg",editors:[{id:"161402",title:"Dr.",name:"Hasan",middleName:null,surname:"Sözen",slug:"hasan-sozen",fullName:"Hasan Sözen"}],productType:{id:"1",chapterContentType:"chapter"}},{type:"book",id:"7955",title:"Advances in Hematologic Malignancies",subtitle:null,isOpenForSubmission:!1,hash:"59ca1b09447fab4717a93e099f646d28",slug:"advances-in-hematologic-malignancies",bookSignature:"Gamal Abdul Hamid",coverURL:"https://cdn.intechopen.com/books/images_new/7955.jpg",editors:[{id:"36487",title:"Prof.",name:"Gamal",middleName:null,surname:"Abdul Hamid",slug:"gamal-abdul-hamid",fullName:"Gamal Abdul Hamid"}],productType:{id:"1",chapterContentType:"chapter"}},{type:"book",id:"7701",title:"Assistive and Rehabilitation Engineering",subtitle:null,isOpenForSubmission:!1,hash:"4191b744b8af3b17d9a80026dcb0617f",slug:"assistive-and-rehabilitation-engineering",bookSignature:"Yves Rybarczyk",coverURL:"https://cdn.intechopen.com/books/images_new/7701.jpg",editors:[{id:"72920",title:"Prof.",name:"Yves",middleName:"Philippe",surname:"Rybarczyk",slug:"yves-rybarczyk",fullName:"Yves Rybarczyk"}],productType:{id:"1",chapterContentType:"chapter"}},{type:"book",id:"7837",title:"Geriatric Medicine and Gerontology",subtitle:null,isOpenForSubmission:!1,hash:"e277d005b23536bcd9f8550046101979",slug:"geriatric-medicine-and-gerontology",bookSignature:"Edward T. Zawada Jr.",coverURL:"https://cdn.intechopen.com/books/images_new/7837.jpg",editors:[{id:"16344",title:"Dr.",name:"Edward T.",middleName:null,surname:"Zawada Jr.",slug:"edward-t.-zawada-jr.",fullName:"Edward T. Zawada Jr."}],productType:{id:"1",chapterContentType:"chapter"}},{type:"book",id:"7123",title:"Current Topics in Neglected Tropical Diseases",subtitle:null,isOpenForSubmission:!1,hash:"61c627da05b2ace83056d11357bdf361",slug:"current-topics-in-neglected-tropical-diseases",bookSignature:"Alfonso J. Rodriguez-Morales",coverURL:"https://cdn.intechopen.com/books/images_new/7123.jpg",editors:[{id:"131400",title:"Dr.",name:"Alfonso J.",middleName:null,surname:"Rodriguez-Morales",slug:"alfonso-j.-rodriguez-morales",fullName:"Alfonso J. Rodriguez-Morales"}],productType:{id:"1",chapterContentType:"chapter"}},{type:"book",id:"7610",title:"Renewable and Sustainable Composites",subtitle:null,isOpenForSubmission:!1,hash:"c2de26c3d329c54f093dc3f05417500a",slug:"renewable-and-sustainable-composites",bookSignature:"António B. Pereira and Fábio A. O. Fernandes",coverURL:"https://cdn.intechopen.com/books/images_new/7610.jpg",editors:[{id:"211131",title:"Prof.",name:"António",middleName:"Bastos",surname:"Pereira",slug:"antonio-pereira",fullName:"António Pereira"}],productType:{id:"1",chapterContentType:"chapter"}},{type:"book",id:"8416",title:"Non-Equilibrium Particle Dynamics",subtitle:null,isOpenForSubmission:!1,hash:"2c3add7639dcd1cb442cb4313ea64e3a",slug:"non-equilibrium-particle-dynamics",bookSignature:"Albert S. Kim",coverURL:"https://cdn.intechopen.com/books/images_new/8416.jpg",editors:[{id:"21045",title:"Prof.",name:"Albert S.",middleName:null,surname:"Kim",slug:"albert-s.-kim",fullName:"Albert S. Kim"}],productType:{id:"1",chapterContentType:"chapter"}},{type:"book",id:"8463",title:"Pediatric Surgery, Flowcharts and Clinical Algorithms",subtitle:null,isOpenForSubmission:!1,hash:"23f39beea4d557b0ae424e2eaf82bf5e",slug:"pediatric-surgery-flowcharts-and-clinical-algorithms",bookSignature:"Sameh Shehata",coverURL:"https://cdn.intechopen.com/books/images_new/8463.jpg",editors:[{id:"37518",title:"Prof.",name:"Sameh",middleName:null,surname:"Shehata",slug:"sameh-shehata",fullName:"Sameh Shehata"}],productType:{id:"1",chapterContentType:"chapter"}}],offset:12,limit:12,total:4392},hotBookTopics:{hotBooks:[],offset:0,limit:12,total:null},publish:{},publishingProposal:{success:null,errors:{}},books:{featuredBooks:[{type:"book",id:"7878",title:"Advances in Extracorporeal Membrane Oxygenation",subtitle:"Volume 3",isOpenForSubmission:!1,hash:"f95bf990273d08098a00f9a1c2403cbe",slug:"advances-in-extracorporeal-membrane-oxygenation-volume-3",bookSignature:"Michael S. Firstenberg",coverURL:"https://cdn.intechopen.com/books/images_new/7878.jpg",editors:[{id:"64343",title:null,name:"Michael S.",middleName:"S",surname:"Firstenberg",slug:"michael-s.-firstenberg",fullName:"Michael S. Firstenberg"}],productType:{id:"1",chapterContentType:"chapter"}},{type:"book",id:"7614",title:"Fourier Transforms",subtitle:"Century of Digitalization and Increasing Expectations",isOpenForSubmission:!1,hash:"ff3501657ae983a3b42fef1f7058ac91",slug:"fourier-transforms-century-of-digitalization-and-increasing-expectations",bookSignature:"Goran S. Nikoli? and Dragana Z. Markovi?-Nikoli?",coverURL:"https://cdn.intechopen.com/books/images_new/7614.jpg",editors:[{id:"23261",title:"Prof.",name:"Goran",middleName:"S.",surname:"Nikolic",slug:"goran-nikolic",fullName:"Goran Nikolic"}],productType:{id:"1",chapterContentType:"chapter"}},{type:"book",id:"8299",title:"Timber Buildings and Sustainability",subtitle:null,isOpenForSubmission:!1,hash:"bccf2891cec38ed041724131aa34c25a",slug:"timber-buildings-and-sustainability",bookSignature:"Giovanna Concu",coverURL:"https://cdn.intechopen.com/books/images_new/8299.jpg",editors:[{id:"108709",title:"Dr.",name:"Giovanna",middleName:null,surname:"Concu",slug:"giovanna-concu",fullName:"Giovanna Concu"}],productType:{id:"1",chapterContentType:"chapter"}},{type:"book",id:"7062",title:"Rhinosinusitis",subtitle:null,isOpenForSubmission:!1,hash:"14ed95e155b1e57a61827ca30b579d09",slug:"rhinosinusitis",bookSignature:"Balwant Singh Gendeh and Mirjana Turkalj",coverURL:"https://cdn.intechopen.com/books/images_new/7062.jpg",editors:[{id:"67669",title:"Prof.",name:"Balwant Singh",middleName:null,surname:"Gendeh",slug:"balwant-singh-gendeh",fullName:"Balwant Singh Gendeh"}],productType:{id:"1",chapterContentType:"chapter"}},{type:"book",id:"7087",title:"Tendons",subtitle:null,isOpenForSubmission:!1,hash:"786abac0445c102d1399a1e727a2db7f",slug:"tendons",bookSignature:"Hasan Sözen",coverURL:"https://cdn.intechopen.com/books/images_new/7087.jpg",editors:[{id:"161402",title:"Dr.",name:"Hasan",middleName:null,surname:"Sözen",slug:"hasan-sozen",fullName:"Hasan Sözen"}],productType:{id:"1",chapterContentType:"chapter"}},{type:"book",id:"7955",title:"Advances in Hematologic Malignancies",subtitle:null,isOpenForSubmission:!1,hash:"59ca1b09447fab4717a93e099f646d28",slug:"advances-in-hematologic-malignancies",bookSignature:"Gamal Abdul Hamid",coverURL:"https://cdn.intechopen.com/books/images_new/7955.jpg",editors:[{id:"36487",title:"Prof.",name:"Gamal",middleName:null,surname:"Abdul Hamid",slug:"gamal-abdul-hamid",fullName:"Gamal Abdul Hamid"}],productType:{id:"1",chapterContentType:"chapter"}},{type:"book",id:"7701",title:"Assistive and Rehabilitation Engineering",subtitle:null,isOpenForSubmission:!1,hash:"4191b744b8af3b17d9a80026dcb0617f",slug:"assistive-and-rehabilitation-engineering",bookSignature:"Yves Rybarczyk",coverURL:"https://cdn.intechopen.com/books/images_new/7701.jpg",editors:[{id:"72920",title:"Prof.",name:"Yves",middleName:"Philippe",surname:"Rybarczyk",slug:"yves-rybarczyk",fullName:"Yves Rybarczyk"}],productType:{id:"1",chapterContentType:"chapter"}},{type:"book",id:"7837",title:"Geriatric Medicine and Gerontology",subtitle:null,isOpenForSubmission:!1,hash:"e277d005b23536bcd9f8550046101979",slug:"geriatric-medicine-and-gerontology",bookSignature:"Edward T. Zawada Jr.",coverURL:"https://cdn.intechopen.com/books/images_new/7837.jpg",editors:[{id:"16344",title:"Dr.",name:"Edward T.",middleName:null,surname:"Zawada Jr.",slug:"edward-t.-zawada-jr.",fullName:"Edward T. Zawada Jr."}],productType:{id:"1",chapterContentType:"chapter"}},{type:"book",id:"7123",title:"Current Topics in Neglected Tropical Diseases",subtitle:null,isOpenForSubmission:!1,hash:"61c627da05b2ace83056d11357bdf361",slug:"current-topics-in-neglected-tropical-diseases",bookSignature:"Alfonso J. Rodriguez-Morales",coverURL:"https://cdn.intechopen.com/books/images_new/7123.jpg",editors:[{id:"131400",title:"Dr.",name:"Alfonso J.",middleName:null,surname:"Rodriguez-Morales",slug:"alfonso-j.-rodriguez-morales",fullName:"Alfonso J. Rodriguez-Morales"}],productType:{id:"1",chapterContentType:"chapter"}},{type:"book",id:"7610",title:"Renewable and Sustainable Composites",subtitle:null,isOpenForSubmission:!1,hash:"c2de26c3d329c54f093dc3f05417500a",slug:"renewable-and-sustainable-composites",bookSignature:"António B. Pereira and Fábio A. O. Fernandes",coverURL:"https://cdn.intechopen.com/books/images_new/7610.jpg",editors:[{id:"211131",title:"Prof.",name:"António",middleName:"Bastos",surname:"Pereira",slug:"antonio-pereira",fullName:"António Pereira"}],productType:{id:"1",chapterContentType:"chapter"}}],latestBooks:[{type:"book",id:"8463",title:"Pediatric Surgery, Flowcharts and Clinical Algorithms",subtitle:null,isOpenForSubmission:!1,hash:"23f39beea4d557b0ae424e2eaf82bf5e",slug:"pediatric-surgery-flowcharts-and-clinical-algorithms",bookSignature:"Sameh Shehata",coverURL:"https://cdn.intechopen.com/books/images_new/8463.jpg",editedByType:"Edited by",editors:[{id:"37518",title:"Prof.",name:"Sameh",middleName:null,surname:"Shehata",slug:"sameh-shehata",fullName:"Sameh Shehata"}],productType:{id:"1",chapterContentType:"chapter",authoredCaption:"Edited by"}},{type:"book",id:"7187",title:"Osteosarcoma",subtitle:"Diagnosis, Mechanisms, and Translational Developments",isOpenForSubmission:!1,hash:"89096359b754beb806eca4c6d8aacaba",slug:"osteosarcoma-diagnosis-mechanisms-and-translational-developments",bookSignature:"Matthew Gregory Cable and Robert Lawrence Randall",coverURL:"https://cdn.intechopen.com/books/images_new/7187.jpg",editedByType:"Edited by",editors:[{id:"265693",title:"Dr.",name:"Matthew Gregory",middleName:null,surname:"Cable",slug:"matthew-gregory-cable",fullName:"Matthew Gregory Cable"}],productType:{id:"1",chapterContentType:"chapter",authoredCaption:"Edited by"}},{type:"book",id:"7955",title:"Advances in Hematologic Malignancies",subtitle:null,isOpenForSubmission:!1,hash:"59ca1b09447fab4717a93e099f646d28",slug:"advances-in-hematologic-malignancies",bookSignature:"Gamal Abdul Hamid",coverURL:"https://cdn.intechopen.com/books/images_new/7955.jpg",editedByType:"Edited by",editors:[{id:"36487",title:"Prof.",name:"Gamal",middleName:null,surname:"Abdul Hamid",slug:"gamal-abdul-hamid",fullName:"Gamal Abdul Hamid"}],productType:{id:"1",chapterContentType:"chapter",authoredCaption:"Edited by"}},{type:"book",id:"7701",title:"Assistive and Rehabilitation Engineering",subtitle:null,isOpenForSubmission:!1,hash:"4191b744b8af3b17d9a80026dcb0617f",slug:"assistive-and-rehabilitation-engineering",bookSignature:"Yves Rybarczyk",coverURL:"https://cdn.intechopen.com/books/images_new/7701.jpg",editedByType:"Edited by",editors:[{id:"72920",title:"Prof.",name:"Yves",middleName:"Philippe",surname:"Rybarczyk",slug:"yves-rybarczyk",fullName:"Yves Rybarczyk"}],productType:{id:"1",chapterContentType:"chapter",authoredCaption:"Edited by"}},{type:"book",id:"7726",title:"Swarm Intelligence",subtitle:"Recent Advances, New Perspectives and Applications",isOpenForSubmission:!1,hash:"e7ea7e74ce7a7a8e5359629e07c68d31",slug:"swarm-intelligence-recent-advances-new-perspectives-and-applications",bookSignature:"Javier Del Ser, Esther Villar and Eneko Osaba",coverURL:"https://cdn.intechopen.com/books/images_new/7726.jpg",editedByType:"Edited by",editors:[{id:"49813",title:"Dr.",name:"Javier",middleName:null,surname:"Del Ser",slug:"javier-del-ser",fullName:"Javier Del Ser"}],productType:{id:"1",chapterContentType:"chapter",authoredCaption:"Edited by"}},{type:"book",id:"8256",title:"Distillation",subtitle:"Modelling, Simulation and Optimization",isOpenForSubmission:!1,hash:"c76af109f83e14d915e5cb3949ae8b80",slug:"distillation-modelling-simulation-and-optimization",bookSignature:"Vilmar Steffen",coverURL:"https://cdn.intechopen.com/books/images_new/8256.jpg",editedByType:"Edited by",editors:[{id:"189035",title:"Dr.",name:"Vilmar",middleName:null,surname:"Steffen",slug:"vilmar-steffen",fullName:"Vilmar Steffen"}],productType:{id:"1",chapterContentType:"chapter",authoredCaption:"Edited by"}},{type:"book",id:"7240",title:"Growing and Handling of Bacterial Cultures",subtitle:null,isOpenForSubmission:!1,hash:"a76c3ef7718c0b72d0128817cdcbe6e3",slug:"growing-and-handling-of-bacterial-cultures",bookSignature:"Madhusmita Mishra",coverURL:"https://cdn.intechopen.com/books/images_new/7240.jpg",editedByType:"Edited by",editors:[{id:"204267",title:"Dr.",name:"Madhusmita",middleName:null,surname:"Mishra",slug:"madhusmita-mishra",fullName:"Madhusmita Mishra"}],productType:{id:"1",chapterContentType:"chapter",authoredCaption:"Edited by"}},{type:"book",id:"8299",title:"Timber Buildings and Sustainability",subtitle:null,isOpenForSubmission:!1,hash:"bccf2891cec38ed041724131aa34c25a",slug:"timber-buildings-and-sustainability",bookSignature:"Giovanna Concu",coverURL:"https://cdn.intechopen.com/books/images_new/8299.jpg",editedByType:"Edited by",editors:[{id:"108709",title:"Dr.",name:"Giovanna",middleName:null,surname:"Concu",slug:"giovanna-concu",fullName:"Giovanna Concu"}],productType:{id:"1",chapterContentType:"chapter",authoredCaption:"Edited by"}},{type:"book",id:"7062",title:"Rhinosinusitis",subtitle:null,isOpenForSubmission:!1,hash:"14ed95e155b1e57a61827ca30b579d09",slug:"rhinosinusitis",bookSignature:"Balwant Singh Gendeh and Mirjana Turkalj",coverURL:"https://cdn.intechopen.com/books/images_new/7062.jpg",editedByType:"Edited by",editors:[{id:"67669",title:"Prof.",name:"Balwant Singh",middleName:null,surname:"Gendeh",slug:"balwant-singh-gendeh",fullName:"Balwant Singh Gendeh"}],productType:{id:"1",chapterContentType:"chapter",authoredCaption:"Edited by"}},{type:"book",id:"7837",title:"Geriatric Medicine and Gerontology",subtitle:null,isOpenForSubmission:!1,hash:"e277d005b23536bcd9f8550046101979",slug:"geriatric-medicine-and-gerontology",bookSignature:"Edward T. Zawada Jr.",coverURL:"https://cdn.intechopen.com/books/images_new/7837.jpg",editedByType:"Edited by",editors:[{id:"16344",title:"Dr.",name:"Edward T.",middleName:null,surname:"Zawada Jr.",slug:"edward-t.-zawada-jr.",fullName:"Edward T. Zawada Jr."}],productType:{id:"1",chapterContentType:"chapter",authoredCaption:"Edited by"}}]},subject:{topic:{id:"757",title:"Semiotics",slug:"electrical-and-electronic-engineering-semiotics",parent:{title:"Electrical and Electronic Engineering",slug:"electrical-and-electronic-engineering"},numberOfBooks:1,numberOfAuthorsAndEditors:19,numberOfWosCitations:3,numberOfCrossrefCitations:17,numberOfDimensionsCitations:27,videoUrl:null,fallbackUrl:null,description:null},booksByTopicFilter:{topicSlug:"electrical-and-electronic-engineering-semiotics",sort:"-publishedDate",limit:12,offset:0},booksByTopicCollection:[{type:"book",id:"3240",title:"Soundscape Semiotics",subtitle:"Localization and Categorization",isOpenForSubmission:!1,hash:"2591f42f72b6e851e6cd65911ec93cb7",slug:"soundscape-semiotics-localisation-and-categorisation",bookSignature:"Herve Glotin",coverURL:"https://cdn.intechopen.com/books/images_new/3240.jpg",editedByType:"Edited by",editors:[{id:"23522",title:"Dr.",name:"Hervé",middleName:null,surname:"Glotin",slug:"herve-glotin",fullName:"Hervé Glotin"}],productType:{id:"1",chapterContentType:"chapter",authoredCaption:"Edited by"}}],booksByTopicTotal:1,mostCitedChapters:[{id:"45589",doi:"10.5772/56872",title:"Clusterized Mel Filter Cepstral Coefficients and Support Vector Machines for Bird Song Identification",slug:"clusterized-mel-filter-cepstral-coefficients-and-support-vector-machines-for-bird-song-identificatio",totalDownloads:1194,totalCrossrefCites:7,totalDimensionsCites:13,book:{slug:"soundscape-semiotics-localisation-and-categorisation",title:"Soundscape Semiotics",fullTitle:"Soundscape Semiotics - Localization and Categorization"},signatures:"Olivier Dufour, Thierry Artieres, Hervé Glotin and Pascale Giraudet",authors:[{id:"23523",title:"Prof.",name:"Pascale",middleName:null,surname:"Giraudet",slug:"pascale-giraudet",fullName:"Pascale Giraudet"},{id:"169025",title:"Dr.",name:"Olivier",middleName:null,surname:"Dufour",slug:"olivier-dufour",fullName:"Olivier Dufour"},{id:"169026",title:"Dr.",name:"Thierry",middleName:null,surname:"Artieres",slug:"thierry-artieres",fullName:"Thierry Artieres"},{id:"169027",title:"Dr.",name:"Hervé",middleName:null,surname:"Glotin",slug:"herve-glotin",fullName:"Hervé Glotin"}]},{id:"45262",doi:"10.5772/56040",title:"Automatic Identification and Interpretation of Animal Sounds, Application to Livestock Production Optimisation",slug:"automatic-identification-and-interpretation-of-animal-sounds-application-to-livestock-production-opt",totalDownloads:1353,totalCrossrefCites:3,totalDimensionsCites:7,book:{slug:"soundscape-semiotics-localisation-and-categorisation",title:"Soundscape Semiotics",fullTitle:"Soundscape Semiotics - Localization and Categorization"},signatures:"Vasileios Exadaktylos, Mitchell Silva and Daniel Berckmans",authors:[{id:"20140",title:"Dr.",name:"Daniel",middleName:null,surname:"Berckmans",slug:"daniel-berckmans",fullName:"Daniel Berckmans"},{id:"20246",title:"Dr.",name:"Vasileios",middleName:null,surname:"Exadaktylos",slug:"vasileios-exadaktylos",fullName:"Vasileios Exadaktylos"},{id:"20391",title:"Mr.",name:"Mitchell",middleName:null,surname:"Silva",slug:"mitchell-silva",fullName:"Mitchell Silva"}]},{id:"45612",doi:"10.5772/56907",title:"Head-Related Transfer Functions and Virtual Auditory Display",slug:"head-related-transfer-functions-and-virtual-auditory-display",totalDownloads:1639,totalCrossrefCites:5,totalDimensionsCites:4,book:{slug:"soundscape-semiotics-localisation-and-categorisation",title:"Soundscape Semiotics",fullTitle:"Soundscape Semiotics - Localization and Categorization"},signatures:"Xiao-li Zhong and Bo-sun Xie",authors:[{id:"155754",title:"Prof.",name:"Bo-Sun",middleName:null,surname:"Xie",slug:"bo-sun-xie",fullName:"Bo-Sun Xie"},{id:"155755",title:"Associate Prof.",name:"Xiao-Li",middleName:null,surname:"Zhong",slug:"xiao-li-zhong",fullName:"Xiao-Li Zhong"}]}],mostDownloadedChaptersLast30Days:[{id:"45612",title:"Head-Related Transfer Functions and Virtual Auditory Display",slug:"head-related-transfer-functions-and-virtual-auditory-display",totalDownloads:1639,totalCrossrefCites:5,totalDimensionsCites:4,book:{slug:"soundscape-semiotics-localisation-and-categorisation",title:"Soundscape Semiotics",fullTitle:"Soundscape Semiotics - Localization and Categorization"},signatures:"Xiao-li Zhong and Bo-sun Xie",authors:[{id:"155754",title:"Prof.",name:"Bo-Sun",middleName:null,surname:"Xie",slug:"bo-sun-xie",fullName:"Bo-Sun Xie"},{id:"155755",title:"Associate Prof.",name:"Xiao-Li",middleName:null,surname:"Zhong",slug:"xiao-li-zhong",fullName:"Xiao-Li Zhong"}]},{id:"45338",title:"Contribution of Precisely Apparent Source Width to Auditory Spaciousness",slug:"contribution-of-precisely-apparent-source-width-to-auditory-spaciousness",totalDownloads:1147,totalCrossrefCites:0,totalDimensionsCites:0,book:{slug:"soundscape-semiotics-localisation-and-categorisation",title:"Soundscape Semiotics",fullTitle:"Soundscape Semiotics - Localization and Categorization"},signatures:"Chiung Yao Chen",authors:[{id:"154975",title:"Prof.",name:"Chiung Yao",middleName:null,surname:"Chen",slug:"chiung-yao-chen",fullName:"Chiung Yao Chen"}]},{id:"45262",title:"Automatic Identification and Interpretation of Animal Sounds, Application to Livestock Production Optimisation",slug:"automatic-identification-and-interpretation-of-animal-sounds-application-to-livestock-production-opt",totalDownloads:1353,totalCrossrefCites:3,totalDimensionsCites:7,book:{slug:"soundscape-semiotics-localisation-and-categorisation",title:"Soundscape Semiotics",fullTitle:"Soundscape Semiotics - Localization and Categorization"},signatures:"Vasileios Exadaktylos, Mitchell Silva and Daniel Berckmans",authors:[{id:"20140",title:"Dr.",name:"Daniel",middleName:null,surname:"Berckmans",slug:"daniel-berckmans",fullName:"Daniel Berckmans"},{id:"20246",title:"Dr.",name:"Vasileios",middleName:null,surname:"Exadaktylos",slug:"vasileios-exadaktylos",fullName:"Vasileios Exadaktylos"},{id:"20391",title:"Mr.",name:"Mitchell",middleName:null,surname:"Silva",slug:"mitchell-silva",fullName:"Mitchell Silva"}]},{id:"45589",title:"Clusterized Mel Filter Cepstral Coefficients and Support Vector Machines for Bird Song Identification",slug:"clusterized-mel-filter-cepstral-coefficients-and-support-vector-machines-for-bird-song-identificatio",totalDownloads:1194,totalCrossrefCites:7,totalDimensionsCites:13,book:{slug:"soundscape-semiotics-localisation-and-categorisation",title:"Soundscape Semiotics",fullTitle:"Soundscape Semiotics - Localization and Categorization"},signatures:"Olivier Dufour, Thierry Artieres, Hervé Glotin and Pascale Giraudet",authors:[{id:"23523",title:"Prof.",name:"Pascale",middleName:null,surname:"Giraudet",slug:"pascale-giraudet",fullName:"Pascale Giraudet"},{id:"169025",title:"Dr.",name:"Olivier",middleName:null,surname:"Dufour",slug:"olivier-dufour",fullName:"Olivier Dufour"},{id:"169026",title:"Dr.",name:"Thierry",middleName:null,surname:"Artieres",slug:"thierry-artieres",fullName:"Thierry Artieres"},{id:"169027",title:"Dr.",name:"Hervé",middleName:null,surname:"Glotin",slug:"herve-glotin",fullName:"Hervé Glotin"}]},{id:"43711",title:"Auditory Distance Estimation in an Open Space",slug:"auditory-distance-estimation-in-an-open-space",totalDownloads:1372,totalCrossrefCites:2,totalDimensionsCites:2,book:{slug:"soundscape-semiotics-localisation-and-categorisation",title:"Soundscape Semiotics",fullTitle:"Soundscape Semiotics - Localization and Categorization"},signatures:"Kim Fluitt, Timothy Mermagen and Tomasz Letowski",authors:[{id:"157339",title:"Ms.",name:"Kim",middleName:null,surname:"Fluitt",slug:"kim-fluitt",fullName:"Kim Fluitt"},{id:"157340",title:"Dr.",name:"Tomasz",middleName:null,surname:"Letowski",slug:"tomasz-letowski",fullName:"Tomasz Letowski"}]},{id:"43951",title:"Evaluation of an Active Microphone with a Parabolic Reflection Board for Monaural Sound-Source-Direction Estimation",slug:"evaluation-of-an-active-microphone-with-a-parabolic-reflection-board-for-monaural-sound-source-direc",totalDownloads:814,totalCrossrefCites:0,totalDimensionsCites:0,book:{slug:"soundscape-semiotics-localisation-and-categorisation",title:"Soundscape Semiotics",fullTitle:"Soundscape Semiotics - Localization and Categorization"},signatures:"Tetsuya Takiguchi, Ryoichi Takashima and Yasuo Ariki",authors:[{id:"2093",title:"Dr.",name:"Tetsuya",middleName:null,surname:"Takiguchi",slug:"tetsuya-takiguchi",fullName:"Tetsuya Takiguchi"},{id:"5503",title:"Dr.",name:"Yasuo",middleName:null,surname:"Ariki",slug:"yasuo-ariki",fullName:"Yasuo Ariki"},{id:"17095",title:"Mr.",name:"Ryoichi",middleName:null,surname:"Takashima",slug:"ryoichi-takashima",fullName:"Ryoichi Takashima"}]},{id:"45170",title:"Source Separation and DOA Estimation for Underdetermined Auditory Scene",slug:"source-separation-and-doa-estimation-for-underdetermined-auditory-scene",totalDownloads:1193,totalCrossrefCites:0,totalDimensionsCites:1,book:{slug:"soundscape-semiotics-localisation-and-categorisation",title:"Soundscape Semiotics",fullTitle:"Soundscape Semiotics - Localization and Categorization"},signatures:"Nozomu Hamada and Ning Ding",authors:[{id:"155108",title:"Prof.",name:"Nozomu",middleName:null,surname:"Hamada",slug:"nozomu-hamada",fullName:"Nozomu Hamada"}]},{id:"45332",title:"Application of Iterative Reverse Time Migration Procedure on Transcranial Thermoacoustic Tomography Imaging",slug:"application-of-iterative-reverse-time-migration-procedure-on-transcranial-thermoacoustic-tomography-",totalDownloads:946,totalCrossrefCites:0,totalDimensionsCites:0,book:{slug:"soundscape-semiotics-localisation-and-categorisation",title:"Soundscape Semiotics",fullTitle:"Soundscape Semiotics - Localization and Categorization"},signatures:"Zijian Liu and Lanbo Liu",authors:[{id:"156358",title:"Dr.",name:"Zijian",middleName:null,surname:"Liu",slug:"zijian-liu",fullName:"Zijian Liu"},{id:"157065",title:"Prof.",name:"Lanbo",middleName:null,surname:"Liu",slug:"lanbo-liu",fullName:"Lanbo Liu"}]}],onlineFirstChaptersFilter:{topicSlug:"electrical-and-electronic-engineering-semiotics",limit:3,offset:0},onlineFirstChaptersCollection:[],onlineFirstChaptersTotal:0},preDownload:{success:null,errors:{}},aboutIntechopen:{},privacyPolicy:{},peerReviewing:{},howOpenAccessPublishingWithIntechopenWorks:{},sponsorshipBooks:{sponsorshipBooks:[{type:"book",id:"6837",title:"Lithium-ion Batteries - Thin Film for Energy Materials and Devices",subtitle:null,isOpenForSubmission:!0,hash:"ea7789260b319b9a4b472257f57bfeb5",slug:null,bookSignature:"Prof. Mitsunobu Sato, Dr. Li Lu and Dr. Hiroki Nagai",coverURL:"https://cdn.intechopen.com/books/images_new/6837.jpg",editedByType:null,editors:[{id:"179615",title:"Prof.",name:"Mitsunobu",middleName:null,surname:"Sato",slug:"mitsunobu-sato",fullName:"Mitsunobu Sato"}],productType:{id:"1",chapterContentType:"chapter"}},{type:"book",id:"9423",title:"Applications of Artificial Intelligence in Process Industry Automation, Heat and Power Generation and Smart Manufacturing",subtitle:null,isOpenForSubmission:!0,hash:"10ac8fb0bdbf61044395963028653d21",slug:null,bookSignature:"Prof. Konstantinos G. Kyprianidis and Prof. Erik Dahlquist",coverURL:"https://cdn.intechopen.com/books/images_new/9423.jpg",editedByType:null,editors:[{id:"35868",title:"Prof.",name:"Konstantinos",middleName:"G.",surname:"Kyprianidis",slug:"konstantinos-kyprianidis",fullName:"Konstantinos Kyprianidis"}],productType:{id:"1",chapterContentType:"chapter"}},{type:"book",id:"9428",title:"New Trends in the Use of Artificial Intelligence for the Industry 4.0",subtitle:null,isOpenForSubmission:!0,hash:"9e089eec484ce8e9eb32198c2d8b34ea",slug:null,bookSignature:"Dr. Luis Romeral Martinez, Dr. Roque A. Osornio-Rios and Dr. Miguel Delgado Prieto",coverURL:"https://cdn.intechopen.com/books/images_new/9428.jpg",editedByType:null,editors:[{id:"86501",title:"Dr.",name:"Luis",middleName:null,surname:"Romeral Martinez",slug:"luis-romeral-martinez",fullName:"Luis Romeral Martinez"}],productType:{id:"1",chapterContentType:"chapter"}},{type:"book",id:"10107",title:"Artificial Intelligence in Oncology Drug Discovery & Development",subtitle:null,isOpenForSubmission:!0,hash:"043c178c3668865ab7d35dcb2ceea794",slug:null,bookSignature:"Dr. John Cassidy and Dr. Belle Taylor",coverURL:"https://cdn.intechopen.com/books/images_new/10107.jpg",editedByType:null,editors:[{id:"244455",title:"Dr.",name:"John",middleName:null,surname:"Cassidy",slug:"john-cassidy",fullName:"John Cassidy"}],productType:{id:"1",chapterContentType:"chapter"}},{type:"book",id:"8903",title:"Carbon Based Material for Environmental Protection and Remediation",subtitle:null,isOpenForSubmission:!0,hash:"19da699b370f320eca63ef2ba02f745d",slug:null,bookSignature:"Dr. Mattia Bartoli and Dr. Marco Frediani",coverURL:"https://cdn.intechopen.com/books/images_new/8903.jpg",editedByType:null,editors:[{id:"188999",title:"Dr.",name:"Mattia",middleName:null,surname:"Bartoli",slug:"mattia-bartoli",fullName:"Mattia Bartoli"}],productType:{id:"1",chapterContentType:"chapter"}},{type:"book",id:"10132",title:"Applied Computational Near-surface Geophysics - From Integral and Derivative Formulas to MATLAB Codes",subtitle:null,isOpenForSubmission:!0,hash:"38cdbbb671df620b36ee96af1d9a3a90",slug:null,bookSignature:"Dr. Afshin Aghayan",coverURL:"https://cdn.intechopen.com/books/images_new/10132.jpg",editedByType:null,editors:[{id:"311030",title:"Dr.",name:"Afshin",middleName:null,surname:"Aghayan",slug:"afshin-aghayan",fullName:"Afshin Aghayan"}],productType:{id:"1",chapterContentType:"chapter"}},{type:"book",id:"10110",title:"Advances and Technologies in Building Construction and Structural Analysis",subtitle:null,isOpenForSubmission:!0,hash:"df2ad14bc5588577e8bf0b7ebcdafd9d",slug:null,bookSignature:"Dr. Ali Kaboli and Dr. Sara Shirowzhan",coverURL:"https://cdn.intechopen.com/books/images_new/10110.jpg",editedByType:null,editors:[{id:"309192",title:"Dr.",name:"Ali",middleName:null,surname:"Kaboli",slug:"ali-kaboli",fullName:"Ali Kaboli"}],productType:{id:"1",chapterContentType:"chapter"}},{type:"book",id:"10175",title:"Ethics in Emerging Technologies",subtitle:null,isOpenForSubmission:!0,hash:"9c92da249676e35e2f7476182aa94e84",slug:null,bookSignature:"Prof. Ali Hessami",coverURL:"https://cdn.intechopen.com/books/images_new/10175.jpg",editedByType:null,editors:[{id:"108303",title:"Prof.",name:"Ali",middleName:null,surname:"Hessami",slug:"ali-hessami",fullName:"Ali Hessami"}],productType:{id:"1",chapterContentType:"chapter"}}],offset:8,limit:8,total:16},humansInSpaceProgram:{},teamHumansInSpaceProgram:{},route:{name:"profile.detail",path:"/profiles/66011/masaru-ishizuka",hash:"",query:{},params:{id:"66011",slug:"masaru-ishizuka"},fullPath:"/profiles/66011/masaru-ishizuka",meta:{},from:{name:null,path:"/",hash:"",query:{},params:{},fullPath:"/",meta:{}}}},function(){var m;(m=document.currentScript||document.scripts[document.scripts.length-1]).parentNode.removeChild(m)}()