Open access peer-reviewed chapter

Cyclotomic and Littlewood Polynomials Associated to Algebras

By José-Antonio de la Peña

Submitted: July 20th 2018Reviewed: October 29th 2018Published: April 6th 2019

DOI: 10.5772/intechopen.82309

Downloaded: 565


Let A be a finite dimensional algebra over an algebraically closed field k . Assume A is a basic connected and triangular algebra with n pairwise non-isomorphic simple modules. We consider the Coxeter transformation ϕ A T as the automorphism of the Grothendieck group K 0 A induced by the Auslander-Reiten translation τ in the derived category D b mod A of the module category mod A of finite dimensional left A -modules. In this paper we study the Mahler measure M χ A of the Coxeter polynomial χ A of certain algebras A . We consider in more detail two cases: (a) A is said to be cyclotomic if all eigenvalues of χ A are roots of unity; (b) A is said to be of Littlewood type if all coefficients of χ A are − 1 , 0 or 1 . We find criteria in order that A is of one of those types. In particular, we establish new records according to Mossingshoff’s list of Record Mahler measures of polynomials q with 1 < M q as small as possible, ordered by their number of roots outside the unit circle.


  • finite dimensional algebra
  • coxeter transformation
  • derived category
  • accessible algebra
  • characteristic polynomial
  • cyclotomic polynomial
  • littlewod type

1. Introduction

Assume throughout the paper that Kis an algebraically closed field. We assume that Ais a triangular finite dimensional basic K-algebra, that is, of the form A=KQ/I, where Iis an ideal of the path algebra KQfor Qa quiver without oriented cycles. In particular, Ahas finite global dimension. The Coxeter transformation ϕAis the automorphism of the Grothendieck group K0Ainduced by the Auslander-Reiten translation τin the derived category DbAsee [1]. The characteristic polynomial χAof ϕAis called the Coxeter polynomialχAof A, or simply χAsee [15, 17]. It is a monic self-reciprocal polynomial, therefore it is χA=a0+a1T+a2T2++an2Tn2+an1Tn1+anTnZT, with ai=anifor 0in, and a0=1=an.

Consider the roots λ1,,λnof χA, the so called spectrumof A. There is a number of measures associated to the absolute values λfor λin the spectrum SpecϕAof A. For instance, the spectral radiusof Ais defined as ρA=maxλ:λSpecϕAand the Mahler measureof χAdefined as MχA=max1λ>1λ. Recently, some explorations on the relations of the Mahler measure MχAand properties of the algebra Ahave been initiated.

For a one-point extension A=BN, we show that MχBMχA. The strongest statements and examples will be given for the class of accessible algebras. We say that an algebra Ais accessible fromBif there is a sequence B=B1,B2,,Bs=Aof algebras such that each Bi+1is a one-point extension (resp. coextension) of Bifor some exceptional Bi-module Mi. As a special case, a K-algebra Ais called accessibleif Ais accessible from the one vertex algebra K.

We say that Ais of cyclotomic typeif the eigenvalues of ϕAlie on the unit circle. Many important finite dimensional algebras are known to be of cyclotomic type: hereditary algebras of finite or tame representation type, canonical algebras, some extended canonical algebras and many others. On the other hand, there are well-known classes of algebras with a mixed behavior with respect to cyclotomicity. For instance, in Section 6 below we consider the class of Nakayama algebras. Let Nnrbe the quotient obtained from the linear quiver with nvertices


with relations xr=0. The Nakayama algebras Nn2are easily proven to be of cyclotomic type, while those of the form Nn3are of cyclotomic type as consequence of lengthly considerations in [18]. The case r=4is more representative: Nn4is of cyclotomic type for all 0n100except for n=10,22,30,42,50,62,70,82and 90. Clearly, if Ais of cyclotomic type then TrϕAkn, for k0. We show the following theorem.

Theorem 1:LetMbe an unimodularn×n-matrix. The following are equivalent:

  1. Mis of cyclotomic type;

  2. for every positive integer 0kn, we have TrMkn.

We also consider algebras Aof Littlewood typewhere χAhas all its coefficients in the set 1,0,1. Among other structure results, we prove.

Proposition.The closureP¯of the setPof roots of Littlewood polynomials, equals the setRof roots of Littlewood series.

Our results make use of well established techniques in the representation theory of algebras, as well as results from the theory of polynomialsand transcendental number theory, where Mahler measure has its usual habitat. We stress here the natural context of these investigations on the largely unexplored overlapping area of these important subjects. Hence, rather than a comprehensive study we understand our work as a preliminary exploration where examples are most valuable.


2. Measures for polynomials

2.1 Self-reciprocal polynomials

A polynomial pzof degree nis said to be self-reciprocal if pz=znp1/z. The following table displays the number anof polynomials pof degree n(for small n) with p0non-zero, bnis the number of such polynomials which are additionally self-reciprocal, and cnis the number of those which are self-reciprocal and where p1is the square of an integer.


Indeed, there is an efficient algorithm to determine such polynomials of given degree n, based on a quadratic bound for n4fn2in terms of Euler totient function, fn.

Cyclotomic polynomials Φnand their products are a natural source for self-reciprocal polynomials. Clearly, Φ1z=z1is not self-reciprocal, but all remaining Φn(with n2) are. Hence, exactly the polynomials z12kn2Φnenwith natural numbers kand enare self-reciprocal with spectral radio one and without eigenvalue zero.

It is not a coincidence that in the above tables we have bn=cn+1for neven and bn=cnfor n odd. Indeed, if pis self-reciprocal of odd degree then p1=0, hence pz=z+1qzwhere qis also self-reciprocal.

2.2 Mahler measure

Let Abe a finite dimensional K-algebra with finite global dimension. The Grothendieck groupK0Aof the category modAof finite dimensional (right) A-modules, formed with respect to short exact sequences, is naturally isomorphic to the Grothendieck group of the derived category, formed with respect to exact triangles.

The Coxeter transformation ϕAis the automorphism of the Grothendieck group K0Ainduced by the Auslander-Reiten translation τ. The characteristic polynomial χATof ϕAis called the Coxeter polynomialχATof A, or simply χA. It is a monic self-reciprocal polynomial, therefore it is χAT=a0+a1T+a2T2++an2Tn2+an1Tn1+anTnZT, with ai=anifor 0in, and a0=1=an.

Consider the roots λ1A,,λnAof χA, the so called spectrumof A. In [15], a measure for polynomials was introduced. Namely, the Mahler measureof χAis MχA=max1i=1nλi. By a celebrated result of Kronecker [9], see also [7, Prop. 1.2.1], a monic integral polynomial p, with p00, has Mp=1if and only if pfactorizes as product of cyclotomic polynomials. As observed in [18], Ais of cyclotomic typeif and only if MχA=1, that is, χATfactorizes as product of cyclotomic polynomials.

2.3 Spectral radius one, periodicity

If the spectrum of Alies in the unit disk, then all roots of χAlie on the unit circle, hence Ahas spectral radius ρA=1. Clearly, for fixed degree there are only finitely many monic integral polynomials with this property.

The following finite dimensional algebras are known to produce Coxeter polynomials of spectral radius one:

  1. hereditary algebras of finite or tame representation type;

  2. all canonical algebras;

  3. (some) extended canonical algebras;

  4. generalizing (2), (some) algebras which are derived equivalent to categories of coherent sheaves.

We put vn=1+x+x2++xn1. Note that vnhas degree n1. There are several reasons for this choice: first of all vn1=n, second this normalization yields convincing formulas for the Coxeter polynomials of canonical algebras and hereditary stars, third representing a Coxeter polynomial — for spectral radius one — as a rational function in the vn‘s relates to a Poincaré series, naturally attached to the setting.

Dynkin typeStar symbolv-factorizationCyclotomic factorizationCoxeter number

In the column ‘v-factorization’, we have added some extra terms in the nominator and denominator which obviously cancel.

Inspection of the table shows the following result:

Proposition.Letkbe an algebraically closed field andAbe a connected, hereditaryk-algebra which is representation-finite. Then the Coxeter polynomialχAdeterminesAup to derived equivalence.□

2.4 Triangular algebras

Nearly all algebras considered in this survey are triangular. By definition, a finite dimensional algebra is called triangularif it has triangular matrix shape


where the diagonal entries Aiare skew-fields and the off-diagonal entries Mij, j>i, are Ai,Aj-bimodules. Each triangular algebra has finite global dimension.

Proposition.LetAbe a triangular algebra over an algebraically closed fieldK. ThenχA1is the square of an integer.

Proof.Let Cbe the Cartan matrix of Awith respect to the basis of indecomposable projectives. Since Ais triangular and Kis algebraically closed, we get detC=1, yielding


Hence χA1is the determinant of the skew-symmetric matrix S=CtC. Using the skew-normal form of S, see [16, Theorem IV.1], we obtain S=UtSUfor some UGLnZ, where Sis a block-diagonal matrix whose first block is the zero matrix of a certain size and where the remaining blocks have the shape 0mimi0with integers mi. The claim follows. □

Which self-reciprocal polynomials of spectral radius one are Coxeter polynomials?The answer is not known. If arbitrary base fields are allowed, we conjecture that all self-reciprocal polynomials are realizable as Coxeter polynomials of triangular algebras. Restricting to algebraically closed fields, already the request that χA1is a square discards many self-reciprocal polynomials, for instance the cyclotomic polynomials Φ4, Φ6, Φ8, Φ10. Moreover, the polynomial f=x3+1, which is the Coxeter polynomial of the non simply-laced Dynkin diagram B3, does not appear as the Coxeter polynomial of a triangular algebra over an algebraically closed field, despite of the fact that f1=0is a square. Indeed, the Cartan matrix


yields the Coxeter polynomial f=x3+αx2+αx+1, where α=abca2b2c2+3. The equation a2+b2+c2abc=3of Hurwitz-Markov type does not have an integral solution. (Use that reduction modulo 3only yields the trivial solution in F3.)

2.5 Relationship with graph theory

Given a (non-oriented) graph Δ, its characteristic polynomialκΔis defined as the characteristic polynomial of the adjacency matrix MΔof Δ. Observe that, since MΔis symmetric, all its eigenvalues are real numbers. For general results on graph theory and spectra of graphs see [4].

There are important interactions between the theory of graph spectra and the representation theory of algebras, due to the fact that if Cis the Cartan matrix of A=KΔ, then MΔis determined by the symmetrization C+Ctof C, since MΔ=C+Ct2I. We shall see that information on the spectra of MΔprovides fundamental insights into the spectral analysis of the Coxeter matrix ΦAand the structure of the algebra A.

A fundamental fact for a hereditary algebra A=KΔ, when Δis a bipartite quiver, that is, every vertex is a sink or source, is that SpecΦAS1R+. This was shown as a consequence of the following important identity.

Proposition. [2] LetA=KΔbe a hereditary algebra withΔa bipartite quiver without oriented cycles. ThenχAx2=xnκΔx+x1, wherenis the number of vertices ofΔandκΔis the characteristic polynomial of the underlying graphΔofΔ.

Proof.Since Δis bipartite, we may assume that the first mvertices are sources and the last nmvertices are sinks. Then the adjacency matrix Aof Δand the Cartan matrix Cof A, in the basis of simple modules, take the form: A=N+Nt, C=InN, where


for certain m×m-matrix D. Since N2=0, then C1=In+N. Therefore


The above result is important since it makes the spectral analysis of bipartite quivers and their underlying graphs almost equivalent. Note, however, that the representation theoretic context is much richer, given the categorical context behind the spectral analysis of quivers. The representation theory of bipartite quivers may thus be seen as a categorification of the class of graphs, allowing a bipartite structure.

Constructions in graph theory. Several simple constructions in graph theory provide tools to obtain in practice the characteristic polynomial of a graph. We recall two of them (see [4] for related results):

  1. Assume that ais a vertex in the graph Δwith a unique neighbor band Δ(resp. Δ) is the full subgraph of Δwith vertices Δ0\a(resp. Δ0\ab), then


  • Let Δibe the graph obtained by deleting the vertex iin Δ. Then the first derivative of κΔis given by


  • The above formulas can be used inductively to calculate the characteristic polynomial of trees and other graphs. They immediately imply the following result that will be used often to calculate Coxeter polynomials of algebras.

    Proposition.LetA=KΔbe a bipartite hereditary algebra. The following holds:

    1. Letabe a vertex in the graphΔwith a unique neighborb. Consider the algebrasBandCobtained as quotients ofAmodulo the ideal generated by the verticesaanda,b, respectively. Then


  • The first derivative of the Coxeter polynomial satisfies:


  • where Ai=KΔ\iis an algebra obtained from Aby ‘killing’ a vertex i.

    Proof.Use the corresponding results for graphs and A’Campo’s formula for the algebras Aand its quotients Ai. □

    3. Important classes of algebras

    In this section we give the definitions and main properties of such classes of finite dimensional algebras where information on their spectral properties is available.

    3.1 Hereditary algebras

    Let Abe a finite dimensional K-algebra. For simplicity we assume A=KΔ/Ifor a quiver Δwithout oriented cycles and Ian ideal of the path algebra. The following facts about the Coxeter transformation ΦAof Aare fundamental:

    1. Let S1,,Snbe a complete system of pairwise non-isomorphic simple A-modules, P1,,Pnthe corresponding projective covers and I1,,Inthe injective envelopes. Then ϕAis the automorphism of K0Adefined by ΦAPi=Ii, where Xdenotes the class of a module Xin K0A.

    2. For a hereditary algebra A=KΔ, the spectral radius ρA=ρΦAdetermines the representation type of Ain the following manner:

      1. Ais representation-finite if 1=ρAis not a root of the Coxeter polynomial χA.

      2. Ais tame if 1=ρARootsχA.

      3. Ais wild if 1<ρA. Moreover, if Ais wild connected, Ringel [20] shows that the spectral radius ρAis a simple root of χA. Then Perron-Frobenius theory yields a vector y+K0AZRwith positive coordinates such that ΦAy+=ρAy+. Since χAis self reciprocal, there is a vector yK0AZRwith positive coordinates such that ΦAy=ρA1y. The vectors y+,yplay an important role in the representation theory of A=KΔ, see [5, 17].

    Explicit formulas, special values. We are discussing various instances where an explicit formula for the Coxeter polynomial is known.

    star quivers. Let Abe the path algebra of a hereditary star p1ptwith respect to the standard orientation, see

    Since the Coxeter polynomial χAdoes not depend on the orientation of Awe will denote it by χp1pt. It follows from [11, prop. 9.1] or [2] that


    In particular, we have an explicit formula for the sum of coefficients of χ=χp1ptas follows:


    This special value of χhas a specific mathematical meaning: up to the factor i=1tpithis is just the orbifold-Euler characteristic of a weighted projective line Xof weight type p1pt. Moreover,

    1. χ1>0if and only if the star p1ptis of Dynkin type, correspondingly the algebra Ais representation-finite.

    2. χ1=0if and only if the star p1ptis of extended Dynkin type, correspondingly the algebra Ais of tame (domestic) type.

    3. χ1<0if and only if p1ptis not Dynkin or extended Dynkin, correspondingly the algebra Ais of wild representation type.

    The above deals with all the Dynkin types and with the extended Dynkin diagrams of type D˜n, n4, and E˜n, n=6,7,8. To complete the picture, we also consider the extended Dynkin quivers of type A˜n(n2) restricting, of course, to quivers without oriented cycles. Here, the Coxeter polynomial depends on the orientation: If p(resp. q) denotes the number of arrows in clockwise (resp. anticlockwise) orientation (p,q1,p+q=n+1), that is, the quiver has type Apq, the Coxeter polynomial χis given by


    Hence χ1=0, fitting into the above picture.

    The next table displays the v-factorization of extended Dynkin quivers.

    Extended Dynkin typeStar symbolWeight symbolCoxeter polynomial
    D˜n, n4[2,2,n-2]22n2x12v22vn2

    Remark:As is shown by the above table, proposition 2.3 extends to the tame hereditary case. That is, the Coxeter polynomial of a connected, tame hereditary K-algebra A(remember, Kis algebraically closed) determines the algebra Aup to derived equivalence. This is no longer true for wild hereditary algebras, not even for trees.

    3.2 Canonical algebras

    Canonical algebras were introduced by Ringel [19]. They form a key class to study important features of representation theory. In the form of tubular canonical algebras they provide the standard examples of tame algebras of linear growth. Up to tilting canonical algebras are characterized as the connected K-algebras with a separating exact subcategory or a separating tubular one-parameter family (see [12]). That is, the module category modΛaccepts a separating tubular familyT=TλλP1K, where Tλis a homogeneous tube for all λwith the exception of ttubes Tλ1,,Tλtwith Tλiof rank pi(1it).

    Canonical algebras constitute an instance, where the explicit form of the Coxeter polynomial is known, see [11] or [10].

    Proposition.LetΛbe a canonical algebra with weight and parameter data(p,λ). Then the Coxeter polynomial ofΛis given by


    The Coxeter polynomial therefore only depends on the weight sequence p. Conversely, the Coxeter polynomial determines the weight sequence — up to ordering.

    3.3 Incidence algebras of posets

    Let Xbe a finite partially ordered set (poset). The incidence algebra KXis the K-algebra spanned by elements exyfor the pairs xyin X, with multiplication defined by exyezw=δyzexw. Finite dimensional right modules over KXcan be identified with commutative diagrams of finite dimensional K-vector spaces over the Hasse diagram of X, which is the directed graph whose vertices are the points of X, with an arrow from xto yif x<yand there is no zXwith x<z<y.

    We recollect the basic facts on the Euler form of posets and refer the reader to [6] for details. The algebra KXis of finite global dimension, hence its Euler form is well-defined and non-degenerate. Denote by CX, ΦXthe matrices of the bilinear form and the corresponding Coxeter transformation with respect to the basis of the simple KX-modules.

    The incidence matrix of X, denoted 1X, is the X×Xmatrix defined by 1Xxy=1if xyand otherwise 1Xxy=0. By extending the partial order on Xto a linear order, we can always arrange the elements of Xsuch that the incidence matrix is uni-triangular. In particular, 1Xis invertible over Z. Recall that the Möbius function μX:X×XZis defined by μXxy=1Xxy1.


    b.Letx,yX. ThenΦXxy=z:zxμXyz.

    Proposition.IfXandYare posets, thenCX×Y=CXCYandΦX×Y=ΦXΦY.

    4. Cyclotomic polynomials and polynomials of Littlewood type

    4.1 Cyclotomic polynomials

    We recall some facts about cyclotomic polynomials.

    The n-cyclotomic polynomial ΦnTis inductively defined by the formula


    The Möbius functionis defined as follows:


    A more explicit expression for the cyclotomic polynomials is given by


    for n2, where vn=1+T+T2++Tn1.

    4.2 Hereditary stars

    A path algebraKΔis said to be of Dynkin typeif the underlying graph Δof Δis one of the ADE-series, that is, of type, An,Dn, for some n1or Ek, for k=6,7,8.

    There are various instances where an explicit formula for the Coxeter polynomial is known.

    Let Abe the path algebra of a hereditary star p1ptwith respect to the standard orientation, see [13].

    Since the Coxeter polynomial χAdoes not depend on the orientation of Awe will denote it by χp1pt. It follows that


    In particular, we have an explicit formula for the sum of coefficients of χp1ptas follows:


    4.3 Wild algebras

    Let cbe the real root of the polynomial T3T1, approximately c=1.325. As observed in [21], a wild hereditary algebra Aassociated to a graph Δwithout multiple arrows has spectral radius ρA>cunless Δis one of the following graphs:

    In these cases, for m8


    where μ0=1.176280is the real root of the Coxeter polynomial


    associated to any hereditary algebra whose underlying graph is 2,3,7. Observe that in these cases, the Mahler measure of the algebra equals the spectral radius.

    4.4 Lehmer polynomial

    In 1933, D. H. Lehmer found that the polynomial


    has Mahler measure μ0=1.176280, and he asked if there exist any smaller values exceeding 1. In fact, the polynomial above is the Coxeter polynomial of the hereditary algebra whose underlying graph 2,3,7is depicted below.

    We say that a matrix Mis of Mahler type(resp. strictly Mahler type) if either MM=1or MMμ0(resp. MM>μ0). Earlier this year, Jean-Louis Verger-Gaugry announced a proof of Lehmer’s conjecture, see The key result (Theorem 5.28, p. 122) is a Dobrowolski type minoration of the Mahler Measure Mβ. Experts are still reading the arguments, but there is no conclusive opinion.

    4.5 Happel’s trace formula

    In [8], Happel shows that the trace of the Coxeter matrix can be expressed as follows:


    where HkAdenotes the k-th Hochschild cohomology group. In particular, if the Hochschild cohomology ring HAis trivial, that is, HiA=0for i>0and H0A=K, then TrϕA=1.

    For an algebra Aand a left A-module Nwe call


    the one-point extensionof Aby N. This construction provides an order of vertices to deal with triangular algebras, that is, algebras KQ/I, where Iis an ideal of the path algebra KQfor Qa quiver without oriented cycles.

    4.6 One-point extensions

    Let Bbe an algebra and Ma B-module. Consider the one-point extension A=BN. In [19] it is shown the Coxeter transformations of Aand Bare related by


    where CBis the Cartan matrixof Bwhich satisfies ϕB=CBTCBand nis the class of Nin the Grothendieck group K0B. In case A=BNwith Nan exceptionalmodule, it follows that


    We recall that the Euler quadratic formis defined as qAx=xCAtxt. Assume that A=BMfor an algebra Band an indecomposable module M. In many cases, we get that qAm>0, for mthe dimension vector of M(for instance, if Mis preprojective, or if qAcoincides with the Tits form of A…)

    Proposition.LetAbe an accessible algebra, such thatqAm>0formthe dimension vector ofM, whereA=BMfor certain algebraBand an indecomposable moduleM. Then the following happens:


    b.if TrϕB=1and qBm=1, then TrϕA=1.

    Proof.Assume that A=BMfor an algebra Band an indecomposable module Msuch that qAm>0for mthe dimension vector of M. Then Bis also accessible. By induction hypothesis, TrϕB1. Then


    This shows (a).

    For (b) assume that TrϕB=1and qBm=1, then


    4.7 Strongly accessible algebras

    Theorem:A finite dimensional accessible algebraAthen it is strongly accessible if and only ifTrϕA=1.

    Proof.Assume Ais strongly accessible from A0. Since qAm1, for A=BMa one-point extension of the subcategory Bof Aby the exceptional module M(since then qAm=dimKEndAM). By the Proposition above


    Conversely, assume that TrϕA=1and write A=BMas a one-point extension of the subcategory Bof Aby the module M. We shall prove that Mis exceptional.


    Equality holds and qBm=1, since Mis indecomposable, it follows that the extension ring of Mis trivial. □

    4.8 Stable matrices

    The following statement is Theorem 1 for stable matrices.

    Proposition.SupposeMis a stable unimodularn×n-matrix. LetχM=c0+c1T+c2T2++cn2Tn2+cn1Tn1+cnTnbe its characteristic polynomial.

    Suppose that0<TrMkmforpkp+n1and certain integers1pandm.

    Then0<TrMkmfor all integerspk.

    In particular,Mis of cyclotomic type.

    Proof.Consider the coefficients c0,c1,cnof χM. Since Mis stable then cn=1,cn1<0,cn2>0and the signs alternate until we meet a jwith cjc0<0. Cayley-Hamilton theorem states that χMM=0. Then




    Let c>0be the common value of the trace of this matrix.

    Write n=2m+rfor r=0or 1. Consider the matrices


    so that we get two expressions of Pas positive linear combinations of powers of M.

    Suppose that n=2m+1. By hypothesis we have TrPn. Moreover, since cn=1then


    The claim follows by induction.

    Otherwise, n=2m. The claim follows similarly.□

    4.9 Theorem 1

    Proof of Theorem 1. Observe that M=ϕAis a real unimodular matrix. One implication of the Theorem was shown before. Suppose that TrMknor equivalently, nTrMknfor 0kn. The Proposition above yields that Mis cyclotomic.□

    4.10 Polynomials of Littlewood type

    An integral self-reciprocal polynomial pt=p0+p1t++pn1tn1+pntnis of Littlewood type if every coefficient non-zero pihas modulus 1. A polynomial ptof Littlewood type with all pi0, for i=0,1,,n, is said to be Littlewood.

    Lemma.Ifzis a root of a polynomial of Littlewood type, then


    Proof.Suppose zis a root of a polynomial of Littlewood type. Then


    for some ϵi1,0,1.

    If z<1then 1z+z2++zn<z/1zso z>1/2. Since zis the root of a polynomial of Littlewood type if and only if z1is, then 1/2<z<2.

    Moreover, if z>1, then 1/z<1and 1/2<1/z<2. Hence 1/2<z<2.□

    4.11 Littlewood series

    Definition.A Littlewood seriesis a power series all of whose coefficients are 1,0or 1.

    Let P={zC:zis the root of some Littlewood polynomial }.


    a.Littlewood series converge for z<1.

    b.A point zCwith z<1lies in Pif and only if some Littlewood series vanishes at this point.

    c.A Littlewood polynomial is not a Littlewood series. But any Littlewood polynomial, say pz=a0++adzdyields a Littlewood series having the same roots zwith z<1: indeed, consider the series


    Thus PR, where Ris the set of roots of Littlewood series. We shall show the Proposition at the Introduction.

    Proof.Let Lbe the set of Littlewood series. Then L=1,0,1, so with the product topology it is homeomorphic to the Cantor set. Choose 0<r<1. Let Fbe the space of finite multisets of points zwith z<r, modulo the equivalence relation generated by SSXwhen X=r.

    Claim. Any Littlewood series has finitely many roots in the disc zr. The map f:LFsending a Littlewood series to its multiset of roots in this disc is continuous.

    Since Lis compact, the image of fis closed. From this we can show that R, the set of roots of Littlewood series, is closed. Since Littlewood polynomials are densely included in Land fis continuous, we get that P, the set of roots of Littlewood polynomials, is dense in R. It follows that P¯=R, as we wanted to show.□

    5. An example

    5.1 Construction

    For ma natural number and let n=3+6m. Let Rnbe an algebra formed by ncommutative squares. Consider the one-point extension Am=RnPnwith Pnthe unique indecomposable projective Rn-module of K-dimension 2. Observe that Am(resp. Cn1) is given by the following quiver with n+1vertices and commutative relations (resp. n1vertices and relations):

    We claim:

    1. χAm=Tn+Tn1T3χAm1+T+1, for all n1. As consequence, the algebras Amand Cnare of Littlewood type;

    2. the number of eigenvalues of ϕAmnot lying in the unit disk is at least m;

    3. MχAm8.

    Proof.(a): Consider m1, n=3+6mand the algebra Bn=R3+6msuch that Am=BnPnand the perpendicular category Pnin DbBnis derived equivalent to modCn1where Cn1is a proper quotient of an algebra derived equivalent to R2+6m. Therefore


    We shall calculate χC2+6m. Observe that C2+6mis tilting equivalent to the one-point extension R1+6mP1. Hence

    χC2+6m= T+1χR1+6mTχR6m=T2+6m+T1+6mT3T+1χR1+6m1TχR6m1+T+1=T2+6m+T1+6mT3χC2+6m1+T+1

    which implies

    χAm+1= T+1Tn+6+Tn+5+T+1T3T+1χRnTTn+5+Tn+4+T+1T3TχCn1=Tn+7+Tn+6T3χAm+T+1

    as claimed.

    As consequence of formula (a) we observe the following:

    (a′) LχAm=4m+5.

    (b) By induction, we shall construct polynomials rmrepresenting χAm.

    For m=0, we have χA0=T4+T3+T2+T+1, which is represented by the polynomial r0=T43T2+1.

    Observe that Tn1+1=vnTvn2then Tn+Tn1+T+1=T+1Tn1+1is represented by wn=Tun1un3.

    For n=4+6m, we define rm=wnT3rm1. We verify by induction on mthat rmrepresents χAm:


    For instance.


    which has ξr1=4changes of sign in the sequence of coefficients. According to Descartes rule of signs, r1has at most ξr1=4positive real roots. Since r1represents χA1, then χA1has at most 2ξr1=8roots in the unit circle. That is, χA1has at least 2roots zwith z1.

    We shall prove, by induction, that rmhas at most ξrm=2m+1positive real roots. Indeed, write


    for some polynomial qmof degree n6with signs of its coefficients +++±so that ξqm=2m. Then


    an addition of three polynomials with signs of coefficients given as follows:


    Hence rm+1=Tn+6n+5Tn+4T3qm+1+n+5T2where the polynomial qm+1of degree nhas signs of its coefficients +++±so that ξqm+1=ξqm+2=2m+1. Hence ξrm=2+ξqm=2m+1.

    By the Lemma below, χAmhas at most 4m+1roots in the unit circle. Equivalently, χAmhas at least 4+6m4m+1=2mroots outside the unit circle. Hence χAmhas at least mroots zsatisfying z>1.

    Lemma.Letqbe a polynomial representing the polynomialp. Assumeqaccepts at mostspositive real roots, thenphas at most2sroots in the unit circle.

    Proof.Let μ1,,μsbe the positive real roots of q. Let z=a+ibbe a root of pwith a2+b2=1. Consider w=c+ida complex number with w2=z. Then 0=pz=wnqw+w1where w+w1=c+id+cid=2c. Then 2c=ϵλjfor some ϵ11and 1js. Hence


    can be selected in two different ways.□

    (c) For n=6m+4we have χAm=Tn+Tn1T3χAm1+T+1. Then


    for m2and ξ1=0.

    We observe that ξmis a product of cyclotomic polynomials. Indeed, since ξm1=0we can write


    for m2and σ1=0.

    Recall Φ2s1=Ts1+Ts2+T+1and Φ2sT=ΦsT. Moreover, Φ3pT=ΦpT3, if pis a power of 2. Altogether this yields




    confirming the claim.

    We estimate the Mahler measure of χAm=ξm+1m1T2m+4χA10. Write χAm=fm+gm, where fmis the cyclotomic summand. Observe that Lgm=LχA10=8and apply Lemma (3.4) with Mfm=1to get


    With the help of computer programs we calculate more accurate values of the Mahler measure of some of the above examples:

    No. verticesNo. roots outside unit diskMahler measure

    Comparing with the list of Record Mahler measures by roots outside the unit circlein Mossinghoff’s web page we see:

    1. for the entry 29the Mahler measure is the same in both tables;

    2. the entries 30and 31have a smaller Mahler measure in our table, establishing new records;

    3. the entry 32of our table seems to be new. Further entries could be calculated.


    6. Coefficients of Coxeter polynomials

    6.1 Derived tubular algebras

    There are interesting invariants associated to the Coxeter polynomial of a triangular algebra A=kΔ/I. For instance, the evaluation of the Coxeter polynomial χA1=m2for some integer m. Clearly, this number is a derived invariant. A simple argument yields that m=0in case Δhas an odd number of vertices. In [14], it was shown that for a representation-finite accessible algebra Awith gl.dim A2the invariant χA1equals zero or one. The criterion was applied to show that a canonical algebra is derived equivalent to a representation-finite algebra if and only if it has weight type 2pp+k, where p2and k0. In particular, the tubular canonical algebra of type 3,3,3is not derived equivalent to a representation-finite algebra, while the tubular algebras of type 2,4,4or 2,3,6are.

    6.2 Strong towers

    Recall from [14] that a strong towerT=A0=kA1An=Aof access to Asatisfies that Ai+1=AiMior MiAifor some exceptional module Miin such a way that, in case Ai+1=AiMi(resp. Ai+1=MiAi), the perpendicular category M¯i(resp. M¯i) of Miin modAiis equivalent to modCi1for some accessible algebra Ci1, i=1,,n1. In the extension situation the perpendicular category Mi(resp. Miin the coextension situation) in DbmodAiis equivalent to DbmodCi1and Biis derived equivalent to a one-point (co-)extension of Ci1. An algebra Cias above is called an i-th perpendicular restrictionof the tower T, observe that it is well-defined only up to derived equivalence. We denote by sithe number of connected components of the algebra Ci; in particular, s1=1.

    There are many examples of strongly accessible algebras, that is, algebras derived equivalent to algebras with a strong tower of access. The following are some instances:

    1. A canonical algebra Cof weight p1ptis strongly accessible if and only if t=3, in that case, Cis derived-equivalent to a representation-finite algebra if and only if the weight type does not dominate 3,3,3.

    2. The following sequence of poset algebras defines strong towers of access:

    6.3 Towering numbers

    Consider a strong tower T=A0=kA1An=Aof access to Asuch that Ai+1is an one-point (co)extension of Aiby Miand Ci1the corresponding i-th perpendicular restriction of T. Let Ci1have si1connected components, i=2,,n1. Define the first towering number ofTas the sum sTA=i=1n2si.

    Theorem. LetAbe a strongly accessible algebra withnvertices, then the first towering numbersTA=i=1n2siofTis a derived invariant, that is, depends only on the derived class ofA. It issTA=n1a2, wherea2is the coefficient of the quadratic term in the Coxeter polynomial ofA.

    Proof.Assume A=Anand B=An1such that A=BMfor Man exceptional B-module and let C=Cn2be the algebra such that modCis derived equivalent to the perpendicular category Mformed in DbmodB. Then χAt=1+tχBttχCt. Write χBt=1+t+i=2n3biti+tn2+tn1and χCt=1+i=1n3citi+tn2. By induction hypothesis we may assume that sB=n2b2. Then a2=b2+1c1. Moreover, since Cis a direct sum accessible algebras, then c1=i=0n21idimkHiC=dimkH0C=sn2. Hence a2=n1sBsn2=n1sA.□

    Corollary.LetT=A1=kAn=Abe a strong tower of access toA. LetA=BMforB=An1withMexceptional andCa perpendicular restriction ofBviaM. Consider the Coxeter polynomialsχAt=1+t+a2t2++an2tn2+tn1+tnandχBt=1+t+b2t2++bn3tn3+tn2+tn1, thena2b2, with equality if and only ifCis connected. In particular,a21.

    Proof.First recall that for a connected accessible algebra the linear term of the Coxeter polynomial has coefficient 1. Let χCt=1+c1t+c2t2++cn4tn4+cn3tn3+tn2be the Coxeter polynomial of C. If Cis the direct sum of connected accessible algebras C1,,Cs, then c1=s. Therefore, a2=b2+b1c1=b2s1b2. By induction hypothesis, we get a21.□

    Let Abe the algebra given by the following quiver with relation γβα=0:

    which is derived equivalent to the quiver algebra Bwith the zero relation as depicted in the second diagram. Clearly, A=AM, where Ais a quiver algebra of type A4and Mis an indecomposable module with Mthe category of modules of the disconnected quiver , that is s3A=2. Moreover s2A=s2A=1and sA=4. On the other hand B=NBsuch that Bis not hereditary. A calculation yields s3B=1and s2B=s2B=2, obviously implying that sB=4.

    Some properties of the invariant s:

    1. Let Aand Bbe accessible algebras and Abe accessible from B, then sBsA. Equality holds exactly when A=B.

    2. Let Abe an accessible schurian algebra (that is for every couple of vertices i,j, dimkAij1), then for every convex subcategory Bwe have sBsA.

    6.4 Totally accessible algebras

    An accessible algebra Awith n=2r+r0vertices, and r001, is said to be totally accessibleif there is a family of (not necessarily connected) algebras Cn=A,Cn2,Cn4,,Cr0satisfying:

    1. Ais derived equivalent to A;

    2. for each 0i=n2jn, there is a strong tower Tj=Cj1=kCji=Ciof access to Ci;

    3. Ci2is an i1-th perpendicular restriction of Tj, that is, Ciis a one-point (co)extension of Cji1by a module Ni1and Ci2is a perpendicular restriction of Cji1via Ni1.

    The tower Tjis said to be a j-th derivativeof the tower T0.

    Examplesthat we have encountered of totally accessible algebras are:

    1. Hereditary tree algebras: since for any conneceted hereditary tree algebra Awith at least 3vertices, there is an arrow abwith aa source (or dually a sink) and A=BPbsuch that the perpendicular restriction of Bvia Pbis the algebra hereditary tree algebra Cobtained from Aby deleting the vertices a,b.

    2. Accessible representation-finite algebras Awith gl.dim A2, since then the perpendicular restrictions of any strong tower (which exists by [14]) satisfy the same set of conditions.

    3. Certain canonical algebras: for instance the tame canonical algebra Aof weight type 2,4,4is an extension A=BMof a hereditary algebra Bof extended Dynkin type 2,4,4by a module Min a tube of rank 4, then the perpendicular restriction of Bvia Mis the hereditary algebra Cof extended Dynkin type 3,3,3, see for example [?](10.1). Since Cis totally accessible, so Ais. Moreover sA=8.

    4. Let Abe an accessible algebra of the form A=BMfor an algebra Band an exceptional module Mand let Cthe perpendicular restriction of Bvia M. If Ais totally accessible, then Band Care totally accessible.

    The following results extend some of the features observed in the examples above.

    Proposition.a.Assume thatAis a totally accessible algebra, thenχA101.

    b.Assume that Ais an accessible but not totally accessible algebra with gl.dim A2, then one of the following conditions hold:

    i.for every exceptional B-module such that A=BMand any perpendicular restriction Cof Bvia M, then Cis not accessible;

    ii.there exists a homological epimorphism ϕ:ABsuch that χB1>1.

    Proof.(a): Consider the perpendicular restriction Cof Bvia M, such that χAt=1+tχBttχCt. Therefore χA1=χC1and moreover, Cis totally accessible. Then by induction hypothesis, χA1=χCm1for a totally accessible algebra Cmwith number of vertices m=1or m=2. Clearly, Cmis either k, kkor hereditary of type A2, which yields the desired result.

    (b): Assume Ais an accessible algebra with gl.dim A2and such that for every homological epimorphism ϕ:ABwe have χB101. Let A=BMfor an accessible algebra Band an exceptional B-module Msuch that Cis a perpendicular restriction of Bvia M. Since gl.dim A2then there is a homological epimorphism ACand gl.dim C2. Observe that for every homological epimorphism ψ:BB(resp. ψ:CC) there is a homological epimorphism ϕ:AB(resp. ϕ:AC), hence χB1(resp. χC1) is 0or 1. By induction hypothesis, Bis totally accessible. Moreover if Cis accessible, then the induction hypothesis yields that Cis totally accessible and also Ais totally accessible, a contradiction. Therefore Cis not accessible.□

    7. On the quadratic coefficient of the Coxeter polynomial of a totally accessible algebra

    7.1 Derived algebras of linear type

    Recall that an extended canonicalalgebra of weight type p1ptis a one-point extension of the canonical algebra of weight type p1ptby an indecomposable projective module. As in (1.3), the extended canonical algebras of type p1p2p3is strongly accessible. Moreover, the extended canonical algebra Aof type 3,4,5(with 12 points) has Coxeter polynomial 1+t+t2++t12which is also the Coxeter polynomial of a linear hereditary algebra Hwith 12vertices. Clearly Aand Hare not derived equivalent.

    The following generalizes a result of Happel who considers the case of Coxeter polynomials associated to hereditary algebras [8].

    Theorem 1.LetAbe a totally accessible algebra withnvertices and letχAt=i=0naitibe the Coxeter polynomial ofA. The following are equivalent:

    1. a2=1;

    2. letT=A1=kAn1An=Abe a strong tower of access toAandCithei-th perpendicular restriction ofT, for all1in2, then the algebrasCiare connected;

    3. A is derived equivalent to a quiver algebra of typeAn.

    Proof.(i) (ii): Let T=A1=kAn=Abe a strong tower of access to A. In case each Ciis connected, then sA=n2, that is a2=1. If a2=1, then n2=sTA=i=1n2siwith each si1. (i) (iii): We know that an algebra Aderived equivalent to a quiver algebra of type Anhas χAt=i=0nti, in particular, a2=1. Assume that an accessible algebra Ahas the quadratic coefficient of its Coxeter polynomial a2=1. Let A=BMfor an accessible algebra B=An1and an exceptional module M. Since Bis also totally accessible with a tower T'=A1=kAn1=Bsatisfying (ii), then the quadratic coefficient of the Coxeter polynomial of Bis b2=1and we may assume that Bis derived equivalent to a quiver algebra of type An1. In particular, Bis representation-finite with a preprojective component Psuch that the orbit graph OPτis of type An1(recall that the orbit graph has vertices the τ-orbits in the quiver Pwith Auslander-Reiten translation τand there is an edge between the orbit of Xand the orbit of Yif there is some numbers a,band an irreducible morphism τaXτbY). Observe that for any Xin DbmodAnot in the orbit of M, there is some translation τaXbelonging to M, implying that in case Mτhas two neighbors in the orbit graph then Mis not connected, that is sn2>1and a2=n1sA0, a contradiction. Therefore, Mτhas just one neighbor in OPτ, hence Ais derived of type An.□

    7.2 Theorem 2

    Consider a tower A1,,An=Aof accessible algebras where Ai+1is a one-point (co)extension of Aiby the indecomposable Miand Ciis such that Miis derived equivalent to DbmodCi. Assume that Cij, for 1jsi, are the connected components of the category Ci. Consider the corresponding Coxeter polynomials:


    where clearly, j=1sini,j=ni.

    Lemma.(α) For every1ji2, we haveaji1.

    (αα) For every1ji2, we haveajici,jandajiaji1.

    Proof.We shall check that (α) implies (αα), then we show that (a’) holds by induction on j.

    Indeed, assume that (α) holds and proceed to show (αα) by induction on j. If j=0,1, then aji=1=aiji. Assume that 2ji2and ajici,jand ajiaji1. Then


    Let 0ji2. If j=0,1we have a0i=1=ci,0and a1is1A=ci,1. Moreover a1i=a1i1. Assume (α) holds for j2, then.


    Theorem 2.LetAbe a totally accessible algebra with Coxeter polynomialχAt=1+t+a2t2++an2tn2+tn1+tn, then:

    1. aj1, for every2jn2;

    2. if for some2jn2, we haveaj=1thenAis derived equivalent to a hereditary algebra of typeAn.

    Proof.Keep the notation as in (4.1). Then (a) is the case i=nof the Lemma above.

    We shall prove (b) by induction on nthe number of vertices of A. Let j=2and assume a2=1, then (3.1) implies that Ais derived equivalent to An. Consider now 2<j<n2and assume that aj=1, we get:


    The last inequality due to (a), hence ajn1=1. Induction hypothesis yields that An1is derived equivalent to An1and its Auslander-Reiten quiver consists of a preprojective component P. In particular, a2n1=1, which implies that sn3An1=1, that is, A=An1Mfor some exceptional module Msuch that Mis derived equivalent to modCfor a connected algebra C, that is, sA=n2and by (3.1), A=BMis derived equivalent to a hereditary algebra of type An.□

    7.3 Examples

    If Ais a representation-finite accessible algebra with gl.dim A2, then Ais totally accessible. On the other hand the algebra Bwith quiver:


    and x3=0is representation-finite and accessible (but not gl.dim B2). The Coxeter polynomial of Bis:


    Then observe that the 6-th coefficient is 1but the algebra Bis not derived equivalent to Dynkin type A12.

    8. On the traces of Coxeter matrices

    Let Abe an algebra such that not all roots of χAare roots of unity. By the result of Kronecker [36], not all of the spectrum of Alies in the unit disk. Equivalently, the spectral radius ρA=maxλ:λeigenvalue  ofϕA>1. Arrange the eigenvalues of ϕAso that μ1,μ2,,μnhave absolute values ρA=r1>r2>>rsand multiplicities m1,,ms, respectively. Therefore s2and


    We define the critical powerκAas the minimal ksuch that


    Since r1is a simple eigenvalue of ϕA, then it follows that κAis well defined due to the existence of ksatisfying the following chain of inequalities:


    The following is a reformulation of Theorem 2.

    Theorem.LetAbe an algebra such that not all roots ofχAare roots of unity. We haveκAn.

    Proof.Indeed, suppose that Ais not of cyclotomic type and κA>n, that is, TrϕAknfor all 0kn. Observe that M=ϕAis a unimodular matrix and therefore, Theorem 2 implies that Mis of cyclotomic type, which yields a contradiction.□

    Remark:We consider explicitly the case n=2in the above Theorem. Obviously, the Cartan matrix of Ais of the form


    for some a1. Then ϕAhas the indicated shape. If Ais not cyclotomic, then a3and TrϕA2=a2222>2.

    9. Stability of a real matrix

    9.1 Stability of matrices and the Lyapunov criterion

    Let Mbe a real invertible n×n-matrix with eigenvalues λj=rjeiθj, for some numbers θj02πand j=1,,n. We will say that Mis stable(resp. semi-stable) if the real part Reeiθj=cosθjof the argument of the eigenvalue λjis positive (resp. non-negative), for every j=1,,n. The following is well-known, we sketch a proof for the sake of completeness.

    Proposition.LetMbe a stable (resp. semi-stable)n×n-matrix. Then the characteristic polynomialχM=Tn+an1Tn1++a1T+a0has coefficients satisfying1njaj>0(resp.0), forj=0,1,,n;

    Proof.Observe that 1npTis the product of polynomials Tαwith αRand Tα+Tα=T22αT+α2+β2with 0β,αR. Stability (resp. semi-stability) implies that α<0(resp. α0) above. Therefore, 1npTis product of polynomials with positive coefficients.□

    Remark:In most of the literature the stability concept we use goes by the name of positive stability, while the stabilityname is used also as Hurwitz stability, or Lyapunov stability.

    The system of differential equations


    is said to be stableif for every vector d=d1dn, the solution vt=etMdof the above system has the property that limtvt=0.

    We recall here the celebrated.

    Lyapunov criterion: The system yt=Mytis stable if and only if Mis a stable matrix, equivalently there is a real positive definite matrix Psuch that


    It is not hard to see that given M, the corresponding Pis unique. A proof of the criterion and its equivalence to other stability conditions are considered in [13].

    9.2 Semi-stable powers

    Let μ1,,μnbe the eigenvalues of the real matrix Mwith μj=ρje2πiθjin polar form. Observe that μjk, for j=1,,n, are the eigenvalues of Mkand


    Lemma.For a positive integerk1the following assertions are equivalent:

    a.Mkis a semi-stable matrix;


    Proof.If Mkis a semi-stable matrix, then μk=ρjkcoskθj+isinkθjhas coskθj0. Since Mis a real matrix then TrMk=j=1nρjkcoskθj0. Therefore


    Assume that TrMk=j=1nλjkcoskθj. Since λjkρjkcoskθjfor j=1,,n, adding up, we get


    Hence we have equalities λjkcoskθj=ρjkcoskθjfor j=1,,n. Then Mkis semi-stable.□

    We say that kis a stable power(resp. semi-stable power) of Mif Mkis a stable (resp. semi-stable) matrix.


    10. Nakayama algebras

    10.1 Cyclotomic Nakayama algebras

    As a well-understood example the representation theory of the Nakayama algebrasstands appart. Let Nnrbe the quotient obtained from the linear quiver with nvertices with radical radAof nilpotency index r.

    For instance, for A=N63the Cartan matrix Cand Coxeter matrix ϕare:


    whose characteristic polynomial is cyclotomic as we know from [18] or might be verified calculating TrϕBkn, for 1k72and applying the criterion of Theorem 1. Indeed, for.


    Starting with k=17the sequence of traces repeats cyclically. Therefore, TrχAk6for all 0k. Then N63is of cyclotomic type.

    10.2 An example

    We recall in some length the argument given in [18] for the cyclotomicity of Nn3, for all n1.

    Consider the algebra R2nwith 2nvertices and whose quiver is given as

    with all commutative relations. The corresponding Coxeter polynomial


    is a product of cyclotomic polynomials, therefore χR2nis a cyclotomic polynomial. In fact R2n=AnA2, where Asis the hereditary algebra associated to the linear quiver 12s.

    For 2m+1odd, we consider.

    The following holds for the sequence of algebras Rnand its Coxeter polynomials χRn:

    1. Rnis derived equivalent to Nn3.

    2. χRn=Tn+Tn1T3χRn6+T+1, for all n6;

    3. MχRn=1.

    Observe that the sequence of algebras Rnforms an interlaced tower of algebras, that is, it is a sequence of triangular algebras R1,,Rn, such that Rsis a basic algebra with ssimple modules and, among others, the condition


    is satisfied for s=1,,n1. Moreover, As+1is a one-point extension (or coextension) of an accessible algebra Asby an exceptional As- module Mssuch that the perpendicular category Msformed in the derived category is triangular equivalent to modAs1, for s=m+1,,n1.

    The following was shown in [18]: Consider an interlaced tower of algebrasAm,,Anwithmn2. IfSpecϕAnis contained in the union of the unit circle and the semi-ray of positive real numbers then either allAiare of cyclotomic type orMχAm<MχAn. In the latter case,MχAn<s=mn1MχAs.

    Since we know that MχR2n=1, for all n0, we conclude that MχRn=1, for all n0. That is the Nakayama algebras of the form Nn3are of cyclotomic type.

    10.3 Non-cyclotomic Nakayama algebras

    Calculation of TrϕAkfor A=Nnrand kin intervals, for data sets nrk, yield interesting information. Namely,

    1. Many Nakayama algebras are of cyclotomic type;

    2. Not all Nakayama algebras are of cyclotomic type. The case r=4illustrates this claim:

      Nn4is of cyclotomic type for all 0n100except for n=10,22,30,42,50,62,70,82and90

    3. A canonical algebra Cof weight p1ptis strongly accessible if and only if t=3, in that case, Cis derived-equivalent to a representation-finite algebra if and only if the weight type does not dominate 3,3,3.

    © 2019 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution 3.0 License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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    José-Antonio de la Peña (April 6th 2019). Cyclotomic and Littlewood Polynomials Associated to Algebras, Polynomials - Theory and Application, Cheon Seoung Ryoo, IntechOpen, DOI: 10.5772/intechopen.82309. Available from:

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