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Polynomials with Symmetric Zeros

Written By

Ricardo Vieira

Submitted: August 11th, 2018 Reviewed: November 26th, 2018 Published: March 23rd, 2019

DOI: 10.5772/intechopen.82728

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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.


  • self-inversive polynomials
  • self-reciprocal polynomials
  • Pisot and Salem polynomials
  • Möbius transformations
  • knot theory
  • Bethe equations

1. Introduction

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 Ror to the unit circle S=zC:z=1. 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.


2. Self-conjugate, self-reciprocal, and self-inversive polynomials

We begin with some definitions:

Definition 1.Let pz=p0+p1z++pn1zn1+pnznbe a polynomial of degree nwith complex coefficients. We shall introduce three polynomials, namely the conjugate polynomialp¯z, the reciprocal polynomialpz, and the inversive polynomialpz, which are, respectively, defined in terms of pzas 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 pz:


From these relations, we plainly see that if ζ1,,ζnare the zeros of a complex polynomial pzof degree n, then, the zeros of p¯zare ζ1¯,,ζn¯, the zeros of pzare 1/ζ1,,1/ζn, and finally, the zeros of pzare 1/ζ1¯,,1/ζn¯. Thus, if pzhas kzeros on R, lzeros on the upper half-plane C+=zC:Imz>0, and mzeros in the lower half-plane C=zC:Imz<0so that k+l+m=n, then p¯zwill have the same number kof zeros on R, lzeros in Cand mzeros in C+. Similarly, if pzhas kzeros on S, lzeros inside Sand mzeros outside S, so that k+l+m=n, then both pzas pzwill have the same number kof zeros on S, lzeros outside Sand mzeros inside S.

These properties encourage us to introduce the following classes of polynomials:

Definition 2.A complex polynomial pzis called1 self-conjugate(SC), self-reciprocal(SR), or self-inversive(SI) if, for any zero ζof pz, the complex-conjugate ζ¯, the reciprocal 1/ζ, or the reciprocal of the complex-conjugate 1/ζ¯is also a zero of pz, respectively.

Thus, the zeros of any SC polynomial are all symmetric to the real line R, while the zeros of the any SI polynomial are symmetric to the unit circle S. 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:

Theorem 1.Ifpzis an SC polynomial of odd degree, then it necessarily has at least one zero onR. Similarly, ifpzis an SR or SI polynomial of odd degree, then it necessarily has at least one zero onS.

Proof.From Definition 2 it follows that the number of non-real zeros of an SC polynomial pzcan only occur in (conjugate) pairs; thus, if pzhas odd degree, then at least one zero of it must be real. Similarly, the zeros of pzor pzthat have modulus different from 1can only occur in (inversive or reciprocal) pairs as well; thus, if pzhas odd degree then at least one zero of it must lie on S.□

Theorem 2.The necessary and sufficient condition for a complex polynomialpzto be SC, SR, or SI is that there exists a complex numberωof modulus1so 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 pzis SC. Then, for any zero ζof pzthe complex-conjugate number ζ¯is also a zero of it. Thus, we can write


with ω=pn/pn¯so that ω=pn/pn¯=1. Now, let us suppose that pzis SR. Then, for any zero ζof pz, the reciprocal number 1/ζis also a zero of it; thus,


with ω=1n/ζ1ζn=pn/p0; now, for any zero ζof pz(which is necessarily different from zero if pzis SR), there will be another zero whose value is 1/ζso that ζ1ζn=1, which implies ω=1. The proof for SI polynomials is analogous and will be concealed; it follows that ω=pn/p0¯in this case.□

Now from (1), (2) and (3), we can conclude that the coefficients of an SC, an SR, and an SI polynomial of degree nsatisfy, 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.

There also exist polynomials whose zeros are symmetric with respect to boththe real line Rand the unit circle S. A polynomial pzwith 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 pzare real, which implies that ω=±1. This suggests the following additional definitions:

Definition 3.A real self-reciprocal polynomial pzthat satisfies the relation pz=ωznp1/zwill be called a positive self-reciprocal(PSR) polynomial if ω=1and a negative self-reciprocal(NSR) polynomial if ω=1.

Thus, the coefficients of any PSR polynomial pz=p0++pnznof degree nsatisfy the relations pk=pnkfor 0kn, while the coefficients of any NSR polynomial pzof degree nsatisfy the relations pk=pnkfor 0kn; this last condition implies that the middle coefficient of an NSR polynomial of even degree is always zero.

Some elementary properties of PSR and NSR polynomials are the following: first, notice that, if ζis a zero of any PSR or NSR polynomial pzof degree n4, then the three complex numbers 1/ζ, ζ¯and 1/ζ¯are also zeros of pz. In particular, the number of zeros of such polynomials which are neither in Sor in Ris always a multiple of 4. Besides, any NSR polynomial has z=1as a zero and pz/z1is PSR; further, if pzhas even degree then z=1is also a zero of it and pz/z21is a PSR polynomial of even degree. In a similar way, any PSR polynomial pzof odd degree has z=1as a zero and pz/z+1is 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.

We also mention that any PSR polynomial of even degree (say, n=2m) can be written in the following form:


an expression that is obtained by using the relations pk=p2mk, 0k2m, and gathering the terms of pzwith the same coefficients. Furthermore, the expression Zsz=zs+zsfor any integer scan be written as a polynomial of degree sin the new variable x=z+1/z(the proof follows easily by induction over s); thus, we can write pz=zmqx, where qx=q0++qmxmis such that the coefficients q0,,qmare certain functions of p0,,pm. From this we can state the following:

Theorem 3.Letpzbe a PSR polynomial of even degreen=2m. For each pairζand1/ζof self-reciprocal zeros ofpzthat lie onS, there is a corresponding zeroξof the polynomialqx, as defined above, in the interval22of the real line.

Proof.For each zero ζof pxthat lie on S, write ζ=efor some θR. Thereby, as qx=qz+1/z=pz/zm, it follows that ξ=ζ+1/ζ=2cosθwill be a zero of qx. This shows us that ξis limited to the interval 22of the real line. Finally, notice that the reciprocal zero 1/ζof pzis mapped to the same zero ξof qx.□

Finally, remembering that the Chebyshev polynomials of first kind, Tnz, are defined by the formula Tn12z+z1=12zn+znfor zC, it follows as well that qx, and hence any PSR polynomial, can be written as a linear combination of Chebyshev polynomials:


3. How these polynomials are related to each other?

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 pzis an SR (SI) polynomial, then p¯zwill 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.

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.

It is an easy matter to verify that Mzmaps Ronto Swhile Wzmaps Sonto R. Besides, Mzmaps the upper (lower) half-plane to the interior (exterior) of S, while Wzmaps the interior (exterior) of Sonto the upper (lower) half-plane. Notice that Wzcan be thought as the inverse of Mzin the Riemann sphere C=C, if we further assume that Mi=, M=1, W1=, and W=i.

Given a polynomial pzof degree n, we define two Möbius-transformed polynomials, namely


The following theorem shows us how the zeros of Qzand Tzare related with the zeros of pz:

Theorem 4.Letζ1,,ζndenote the zeros ofpzandη1,,ηnthe respective zeros ofQz. Providedp10, we have thatη1=Wζ1,,ηn=Wζn. Similarly, ifτ1,τnare the zeros ofTz, then we haveτ1=Mζ1,,τn=Mζn, provided thatpi0.

Proof.In fact, inverting the expression for Qzand evaluating it in any zero ζkof pzwe get that pζk=i/2nζk1nQWζk=0for 0kn. Provided that z=1is not a zero of pzwe get that ηk=Wζkis a zero of Qz. The proof for the zeros of Tzis analogous.□

This result also shows that Qzand Tzhave the same degree as pzwhenever p10or pi0, respectively. In fact, if pzhas a zero at z=1of multiplicity mthen Qzwill be a polynomial of degree nm, the same being true for Tzif pzhas a zero of multiplicity mat z=i. This can be explained by the fact that the points z=1and z=iare mapped to infinity by Wzand Mz, respectively.

The following theorem shows that the set of SI polynomials are isomorphic to the set of SC polynomials:

Theorem 5.Letpzbe an SI polynomial. Then, the transformed polynomialQz=z+inpMzis an SC polynomial. Similarly, ifpzis an SC polynomial, thenTz=z1npWzwill be an SI polynomial.

Proof.Let ζand 1/ζ¯be two inversive zeros an SI polynomial pz. Then, according to Theorem 4, the corresponding zeros of Qzwill be:


Thus, any pair of zeros of pzthat are symmetric to the unit circle are mapped in zeros of Qzthat are symmetric to the real line; because pzis SI, it follows that Qzis SC. Conversely, let ζand ζ¯be two zeros of an SC polynomial pz; then the corresponding zeros of Tzwill be:


Thus, any pair of zeros of pzthat are symmetric to the real line are mapped in zeros of Tzthat are symmetric to the unit circle. Because pzis SC, it follows that Tzis SI.□

We can also verify that any SI polynomial with ω=1is mapped to a real polynomial through Mzand any real polynomial is mapped to an SI polynomial with ω=1through Wz. Thus, the set of SI polynomials with ω=1is isomorphic to the set of real polynomials. Besides, an SI polynomial with ω1can be transformed into another one with ω=1by 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.


4. Zeros location theorems

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 Sare 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].

4.1 Polynomials that do not necessarily have symmetric zeros

The following theorems are classics (see [1] for the proofs):

Theorem 6. (Rouché). Letqzandrzbe polynomials such thatqz<rzalong all points ofS. Then, the polynomialpz=qz+rzhas the same number of zeros insideSas the polynomialrz, counted with multiplicity.

Thus, if a complex polynomial pz=p0++pkzk++pnznof degree nis such that pk>p0++pk1+pk+1++pn, then pzwill have exactly kzeros inside S, counted with multiplicity.

Theorem 7. (Gauss and Lucas)The zeros of the derivativepzof a polynomialpzlie all within the convex hull of the zeros of thepz.

Thereby, if a polynomial pzhas all its zeros on S, then all the zeros of pzwill lie in or on S. In particular, the zeros of pzwill lie on Sif, and only if, they are multiple zeros of pz.

Theorem 8. (Cohn)A necessary and sufficient condition for all the zeros of a complex polynomialpzto lie onSis thatpzis SI and that its derivativepzdoes not have any zero outsideS.

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 pz=pnzn++p0will lie on Sif, and only if, pn=p0and all the zeros of pzdo not lie outside S.

Restricting ourselves to polynomials with real coefficients, Eneström and Kakeya [8, 9, 10] independently presented the following theorem:

Theorem 9. (Eneström and Kakeya)Letpzbe a polynomial of degreenwith real coefficients. If its coefficients are such that0<p0p1pn1pn, then all the zeros ofpzlie in or onS. Likewise, if the coefficients ofpzare such that0<pnpn1p1p0, then all the zeros ofpzlie on or outsideS.

The following theorems are relatively more recent. The distribution of the zeros of a complex polynomial regarding the unit circle Swas presented by Marden in [1] and slightly enhanced by Jury in [11]:

Theorem 10. (Marden and Jury)Letpzbe a complex polynomial of degreenandpzits reciprocal. Construct the sequence of polynomialsPjz=k=0njPj,kzksuch thatP0z=pzandPj+1z=pj,0¯Pjzpj,nj¯Pjzfor0jn1so that we have the relationspj+1,k=pj,0¯pj,kpj,njpj,njk¯. Letδjdenote the constant terms of the polynomialsPjz, i.e.,δj=pj,0andΔk=δ1δk. Thus, ifNof the productsΔkare negative andnNof the productsΔkare positive so that none of them are zero, thenpzhasNzeros insideS,nNzeros outsideSand no zero onS. On the other hand, ifΔk0for somek<nbutPk+1z=0, thenpzhas eithernkzeros onSornkzeros symmetric toS. It has additionallyNzeros insideSandkNzeros outsideS.

A simple necessary and sufficient condition for all the zeros of a complex polynomial to lie on Swas presented by Chen in [12]:

Theorem 11. (Chen)A necessary and sufficient condition for all the zeros of a complex polynomialpzof degreento lie onSis that there exists a polynomialqzof degreenmwhose zeros are all in or onSand such thatpz=zmqz+ωqzfor some complex numberωof modulus1.

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 abof the real line is determined by the formula N=varSbvarSa, where varSξmeans the number of sign variations of the Sturm sequence Sxevaluated at x=ξ) 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].

4.2 Real self-reciprocal polynomials

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.

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]:

Theorem 12. (Chen and Chinen)Letpzbe a PSR polynomial of degreenthat is written in the formpz=p0+p1z++pkzk+pkznk+pk1znk+1++p0znand such that0<pk<pk1<<p1<p0. Then all the zeros ofpzare onS.

Going in the same direction, Choo found in [15] the following condition:

Theorem 13. (Choo)Letpzbe a PSR polynomial of degreenand such that its coefficients satisfy the following conditions:npnn1pn1k+1pk+1>0andk+1pk+1j=0kj+1pj+1jpjfor0kn1. Then, all the zeros ofpzare onS.

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 S. One of the main theorems is the following:

Theorem 14. (Lakatos)Letpzbe a PSR polynomial of degreen>2. Ifpnk=1n1pnpk, then all the zeros ofpzlie onS. Moreover, the zeros ofpzare all simple, except when the equality takes place.

For PSR polynomials of odd degree, Lakatos and Losonczi [17] found a stronger version of this result:

Theorem 15. (Lakatos and Losonczi)Letpzbe a PSR polynomial of odd degree, sayn=2m+1. Ifp2m+1cos2ϕmk=12mp2m+1pk, whereϕm=π/4m+1, then all the zeros ofpzlie onS. The zeros are simple except when the equality is strict.

Theorem 14 was generalized further by Lakatos and Losonczi in [18]:

Theorem 16. (Lakatos and Losonczi)All zeros of a PSR polynomialpzof degreen>2lie onSif the following conditions hold:pn+rk=1n1pkpn+r,pnr0, andpnr, forrR.

Other conditions for all the zeros of a PSR polynomial to lie on Swere presented by Kwon in [19]. In its simplest form, Kown’s theorem can be enunciated as follows:

Theorem 17. (Kwon)Letpzbe a PSR polynomial of even degreen2whose leading coefficientpnis positive andp0p1pn. In this case, all the zeros ofpzwill lie onSif, eitherpn/2k=0npkpn/2, orp10andpn12k=1n1pkpn/2.

Modified forms of this theorem hold for PSR polynomials of odd degree and for the case where the coefficients of pzdo not have the ordination above—see [19] for these cases. Kwon also found conditions for all but two zeros of pzto lie on Sin [20], which is relevant to the theory of Salem polynomials—see Section 5.

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 S. Kim and Park [22] and then Kim and Lee [23] presented conditions for which all the zeros of certain PSR polynomials lie on S(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 S. 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 1—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 Polynomialsand 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 10can be solved by radicals.

4.3 Complex self-reciprocal and self-inversive polynomials

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).

Theorem 18. (Cohn)An SI polynomialpzhas as many zeros outsideSas does its derivativepz.

This follows directly from Cohn’s Theorem 8 for the case where pzis SI. Besides, we can also conclude from this that the derivative of pzhas no zeros on Sexcept at the multiple zeros of pz. Furthermore, if an SI polynomial pzof degree nhas exactly kzeros on S, while its derivative has exactly lzeros in or on S, both counted with multiplicity, then n=2l+1k.

O’Hara and Rodriguez [41] showed that the following conditions are always satisfied by SI polynomials whose zeros are all on S:

Theorem 19. (O’Hara and Rodriguez)Letpzbe an SI polynomial of degreenwhose zeros are all onS. Then, the following inequality holds:j=0npj2pz2, wherepzdenotes the maximum modulus ofpzon the unit circle; besides, if this inequality is strict then the zeros ofpzare rotations ofnth roots of unity. Moreover, the following inequalities are also satisfied:ak12pzif kn/2and ak22pzfor k=n/2.

Schinzel in [42], generalized Lakatos Theorem 14 for SI polynomials:

Theorem 20. (Schinzel)Letpzbe an SI polynomial of degreen. If the inequalitypninfa,bC:b=1k=0napkbnkpn, then all the zeros ofpzlie onS. These zero are simple whenever the equality is strict.

In a similar way, Losonczi and Schinzel [43] generalized theorem 15 for the SI case:

Theorem 21. (Losonczi and Schinzel)Letpzbe an SI polynomial of odd degree, i.e.,n=2m+1. Ifp2m+1cos2ϕminfa,bC:b=1k=12m+1apkb2m+1kp2m+1, whereϕm=π/4m+1, then all the zeros ofpzlie onS. The zeros are simple except when the equality is strict.

Another sufficient condition for all the zeros of an SI polynomial to lie on Swas presented by Lakatos and Losonczi in [44]:

Theorem 22. (Lakatos and Losonczi)Letpzbe an SI polynomial of degreenand suppose that the inequalitypn12k=1n1pkholds. Then, all the zeros ofpzlie onS. Moreover, the zeros are all simple except when an equality takes place.

In [45], Lakatos and Losonczi also formulated a theorem that contains as special cases many of the previous results:

Theorem 23. (Lakatos and Losonczi)Letpz=p0++pnznbe an SI polynomial of degreen2anda,b, andcbe complex numbers such thata0,b=1, andc/pnR,0c/pn1. Ifpn+cap0bnpn+k=1n1apkbnkcpn+apnpn, then, all the zeros ofpzlie onS. Moreover, these zeros are simple if the inequality is strict.

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 S:

Theorem 24. (Losonczi)Letpzbe a monic complex SR polynomial of even degree, sayn=2m. Then, all the zeros ofpzwill lie onSif, and only if, there exist real numbersα1,,α2m, all with moduli less than or equal to2, that satisfy the inequalities:pk=1kl=0k/2mk+2llσk2l2mα1α2m,0km, whereσk2mα1α2mdenotes thekth elementary symmetric function in the2mvariablesα1,,α2m.

Losonczi, in [46], also showed that if all the zeros of a complex monic reciprocal polynomial are on S, then its coefficients are all real and satisfy the inequality pnnkfor 0kn.

The theorems above give conditions for allthe zeros of SI or SR polynomials to lie on S. In many cases, however, we need to verify if a polynomial has a given numberof zeros (or none) on the unit circle. Considering this problem, Vieira in [47] found sufficient conditions for an SI polynomial of degree nto have a determined number of zeros on the unit circle. In terms of the length, Lpz=p0++pnof a polynomial pzof degree n, this theorem can be stated as follows:

Theorem 25. (Vieira)Letpzbe an SI polynomial of degreen. If the inequalitypnm14nnmLpz,m<n/2, holds true, thenpzwill have exactlyn2mzeros onS; besides, all these zeros are simple when the inequality is strict. Moreover,pzwill have no zero onSif, forneven andm=n/2, the inequalitypm>12Lpzis satisfied.

The case m=0corresponds to Lakatos and Losonczi Theorem 14 for all the zeros of pzto lie on S. 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.

Other results on the distribution of zeros of SI polynomials include the following: Sinclair and Vaaler [49] showed that a monic SI polynomial pzof degree nsatisfying the inequalities Lrpz2+2rn11ror Lrpz2+2rl21r, where r1, Lrpz=p0r++pnr, and lis the number of non-null terms of pz, has all their zeros on S; the authors also studied the geometry of SI polynomials whose zeros are all on S. Choo and Kim applied Theorem 11 to SI polynomials in [50]. Hypergeometric polynomials with all their zeros on Swere 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 S.


5. Where these polynomials are found?

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.).

5.1 Polynomials with small Mahler measure

Given a monic polynomial pzof degree n, with integer coefficients, the Mahler measure of pz, denoted by Mpz, is defined as the product of the modulus of all those zeros of pzthat lie in the exterior of S[61]. That is


where ζ1,,ζnare the zeros3 of pz. Thus, if a monic integer polynomial pzhas all its zeros in or on the unit circle, we have Mpz=1; in particular, all cyclotomic polynomials (which are PSR polynomials whose zeros are the primitive roots of unity, see [1]) have Mahler measure equal to 1. In a sense, the Mahler measure of a polynomial pzmeasures how close it is to the cyclotomic polynomials. Therefore, it is natural to raise the following:

Problem 1. (Mahler)Find the monic, integer, non-cyclotomic polynomial with the smallest Mahler measure.

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 S, in particular among those with only one zero outside S. A monic integer polynomial that has exactly one zero outside Sis called a Pisot polynomialand its unique zero of modulus greater than 1is called its Pisot number[62]. A breakthrough towards the solution of Mahler’s problem was given by Smyth in [63]:

Theorem 26. (Smyth)The Pisot polynomialSz=z3z1is 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, ζand 1/ζ) not lying on Sis called a Salem polynomial[62, 64]. It can be shown that a Pisot polynomial with at least one zero on Sis 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 sis said to be small if s<σ; up to date, only 47small 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:

Conjecture 1. (Lehmer)The monic integer polynomial with the smallest Mahler measure is the Lehmer polynomialLz=z10+z9z7z6z5z4z3+z+1, a Salem polynomial whose Mahler measure isΛ1.17628081826, known as Lehmer’s constant.

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:

Problem 2. (Lehmer)Answer whether there exists or not a positive numberϵsuch that the Mahler measure of any monic, integer, and non-cyclotomic polynomialpzsatisfies the inequalityMpz>1+ϵ.

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 λis any zero of the aforementioned Lehmer’s polynomial Lz, then,


5.2 Knot theory

A knotis a closed, non-intersecting, one-dimensional curve embedded on R3[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.

Figure 1.

A table of prime knots up to six crossings. In the Alexander-Briggs notation these knots are, in order,01,31,41,51,52,61,62, and63.

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].

What is interesting for us here is that the Alexander polynomials are PSR polynomials of even degree (say, n=2m) and with integer coefficients.5 Thus, they have the following general form:


where δiN, 0im. In Table 1, we present the $\delta_{m - 1}$Alexander polynomials for the prime knots up to six crossings.

KnotAlexander polynomial ΔtKnotAlexander polynomial Δt

Table 1.

Alexander polynomials for prime knots up to six crossings.

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 KnotP2,3,7is nothing but the Lehmer polynomial Lzintroduced 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.

5.3 Bethe equations

Bethe equationswere 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:

xiL=1N1k=1,kiN xixk2Δxi+1xixk2Δxk+1,1iN,E18

where LNis the length of the chain, NNis the excitation number and ΔRis the so-called spectral parameter. A solution of (18) consists in a (non-ordered) set X=x1xNof the unknowns x1,,xNso that (18) is satisfied. Notice that the Bethe equations satisfy the important relation x1Lx2LxNL=1, which suggests an inversive symmetry of their zeros.

In [76], Vieira and Lima-Santos showed that the solutions of (18), for N=2and arbitrary L, are given in terms of the zeros of certain SI polynomials. In fact, (18) becomes a system of two coupled algebraic equations for N=2, namely,


Now, from the relation x1Lx2L=1we can eliminate one of the unknowns in (19)—for instance, by setting x2=ωa/x1, where ωa=exp2πia/L, 1aL, are the roots of unity of degree L. Replacing these values for x2into (19), we obtain the following polynomial equations fixing x1:


We can easily verify that the polynomial pazis SI for each value of a. They also satisfy the relations paz=zLpωa/z, 1aL, which means that the solutions of (19) have the general form X=ζωa/ζfor ζany zero of paz. In [76], the distribution of the zeros of the polynomials pazwas analyzed through an application of Vieira’s Theorem 25. It was shown that the exact behavior of the zeros of the polynomials paz, for each a, depends on two critical values of Δ, namely


as follows: if ΔΔa1, then all the zeros of pazare on S; if ΔΔa2, then all the zeros of pazbut two are on S; (see [76] for the case Δa1<Δ<Δa2and more details).

Finally, we highlight that the polynomial pazbecomes a Salem polynomial for a=Land integer values of Δ. This was one of the first appearances of Salem polynomials in physics.

5.4 Orthogonal polynomials

An infinite sequence P=PnznNof polynomials Pnzof degree nis said to be an orthogonal polynomial sequenceon the interval lrof the real line if there exists a function wx, positive in lrR, such that


where K0, K1, etc. are positive numbers. Orthogonal polynomial sequences on the real line have many interesting and important properties—see [77].

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 T=TnznNof the Möbius-transformed polynomials Tnz=z1nPnWz, where Wz=iz+1/z1, is an SI polynomial sequence with all their zeros on the unit circle S—see Table 2 for an example. We highlight that the polynomials TnzTalso have properties similar to the original polynomials PnzPas, for instance, they satisfy an orthogonality condition on the unit circle and a three-term recurrence relation, their zeros lie all on Sand are simple, for n1the zeros of Tnzinterlaces with those of Tn+1zand so on—see [78, 79] for more details.

Hermite polynomialsMöbius-transformed Hermite polynomials

Table 2.

Hermite and Möbius-transformed Hermite polynomials, up to 4th degree.


6. Conclusions

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.



We thank the editorial staff for all the support during the publishing process and also the Coordination for the Improvement of Higher Education(CAPES).


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  • 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.
  • The zeros of such polynomials present a fractal behavior, as was first discovered by Odlyzko and Poonen in [36].
  • The Mahler measure of a monic integer polynomial p z can also be defined without making reference to its zeros through the formula M p z = exp ∫ 0 1 log p e 2 πit dt —see [61].
  • 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.
  • Alexander polynomials can also be defined as Laurent polynomials, see [70].

Written By

Ricardo Vieira

Submitted: August 11th, 2018 Reviewed: November 26th, 2018 Published: March 23rd, 2019