Cyclotomic factors of Coxeter polynomials for diagrams obtained by decorating $A_n$: a spherical-origin criterion and effective periodicity

For a sequence $F_n$ of Coxeter diagrams obtained by attaching two fixed pointed diagrams $(B,s)$ and $(C,t)$ to the ends of the path $A_n$, Gross, Hironaka and McMullen (Journal of Number Theory, 2009) proved a dichotomy: either every $F_n$ is spherical or affine, or the cyclotomic factor of the Coxeter polynomial $F_n(x)$ is eventually periodic in $n$. For the special family $E_n$ they determined this factor completely (for all $n\ne 9$): the cyclotomic zeros have orders in $\{2,3,5,8,12,18,30\}$, depend only on $n\bmod 360$, and arise exclusively from splittings of the diagram into spherical subdiagrams, that is, from joins at a vertex whose pieces are spherical. The authors left open (i) finding a general condition on the decorations that guarantees this exclusive spherical origin of the cyclotomic factors, and (ii) making the eventual periodicity effective: an explicit threshold and modulus (or a certificate that every member is spherical or affine) together with a terminating verification procedure for general decorated families. Later work applies the paper's Mann-type two-term bound in adjacent settings - graph truncations and subfactor vines, star-like trees, rational surface automorphisms, tree characteristic polynomials - but no general criterion of type (i) and no general effective form of type (ii) for Coxeter polynomials of two-sided $A_n$ decorations are known. Both components are open.

Background

A Coxeter system $(W,S)$ is a group generated by involutions $S$ whose pairwise products have prescribed finite orders $m_{st}\ge 2$; it is encoded by a Coxeter diagram whose vertices are the generators, with an edge of weight $m_{st}-2$ joining $s$ to $t$ whenever $m_{st}>2$ (by convention an unlabeled edge has weight one). The system acts on $\mathbb{R}^S$ preserving a symmetric bilinear form $B$, and the product $w$ of the generators in any order is a Coxeter element; when the diagram is a tree, the conjugacy class of $w$, and hence the characteristic polynomial $F(x)=\det(xI-w)$ (the Coxeter polynomial), is independent of the ordering. For bipartite diagrams one uses the bicolored Coxeter element, again well-defined up to conjugacy. If the diagram has no multiple edges, $W$ preserves a lattice and $F(x)\in\mathbb{Z}[x]$; the statements below assume $F_n(x)\in\mathbb{Z}[x]$ wherever this integrality is needed.

A diagram is called spherical if its Coxeter group is finite (equivalently, $B$ is positive definite - the Dynkin diagrams $A_n$, $D_n$, $E_n$, etc.), affine if $B$ is positive semidefinite with a one-dimensional radical (the extended Dynkin diagrams $\widetilde{A}_n,\dots$), and hyperbolic if $B$ has signature $(n-1,1)$. Write $\Phi_k(x)$ for the $k$-th cyclotomic polynomial, the minimal polynomial over $\mathbb{Q}$ of a primitive $k$-th root of unity. The Coxeter polynomial factors uniquely as $F(x)=C(x)S(x)$, where the cyclotomic factor $C(x)$, the product of all $\Phi_k(x)$ dividing $F(x)$, collects the zeros that are roots of unity. By a classical theorem of A'Campo (1976), when the diagram is connected and hyperbolic the complementary factor $S(x)$ is a Salem polynomial: an irreducible reciprocal polynomial (reciprocal meaning $x^dP(x^{-1})=P(x)$ for a polynomial $P$ of degree $d$) with a unique root $\lambda>1$ outside the unit circle, so that $\lambda$ is a Salem number. The smallest known Salem number, Lehmer's number, $\approx 1.17628$, arises this way from the diagram $E_{10}$.

The Coxeter polynomial of the path $A_n$ is $(x^{n+1}-1)/(x-1)$, so its zeros are all the $(n+1)$-st roots of unity except $1$ - the archetype of a fully cyclotomic family. A pointed Coxeter diagram is a pair $(F,s)$ with a distinguished vertex $s$; deleting $s$ leaves a possibly disconnected diagram, whose Coxeter polynomial - the product of the polynomials of its connected components - is denoted $\widehat{F}(x)$. The join of pointed diagrams $(F_i,s_i)_{i=1}^m$ is obtained by adding a new vertex connected to each $s_i$ by a single edge. Two inputs from the 2009 paper of Gross, Hironaka and McMullen (GHM) govern joins. Their Theorem 1.5: if $F$ is the Coxeter diagram obtained by joining together diagrams $F_1,\dots,F_m$ at a single new vertex, then any zero of two or more of the Coxeter polynomials $F_i(x)$ is also a zero of $F(x)$. And a direct matrix computation (equation (3.2) of the source) expresses the Coxeter polynomial of the join:

$$F(x)=F_1(x)\cdots F_m(x)\left((x+1)-x\sum_{i=1}^m\frac{\widehat{F_i}(x)}{F_i(x)}\right).$$

This is the mechanism by which subdiagrams force cyclotomic factors. The $E_n$ diagram ($n\ge 3$) is the join of $E_i$ and $A_{n-i-1}$ for $3\le i<n-1$: for example, $E_{38}$ is the join of $E_8$ and $A_{29}$, and since $E_8(x)=\Phi_{30}(x)$ divides $A_{29}(x)=(x^{30}-1)/(x-1)$, $E_8(x)$ divides $E_{38}(x)$.

Two further inputs from GHM are needed. First, Mann's theorem on linear relations between roots of unity (1965): in a minimal (primitive) vanishing integer linear combination of roots of unity, the order of the combination divides the product of the primes at most the number of terms. GHM recast it as a two-term bound (their Theorem 2.1): if $\zeta_k^nQ(\zeta_k)+R(\zeta_k)=0$ with $Q,R\in\mathbb{Z}[x]$ not both zero and $\zeta_k$ a primitive $k$-th root of unity, then either $Q(x)=\pm x^iR(x)$ for some $i\in\mathbb{Z}$, or $k\le 2s\max(\deg Q,\deg R)$, where $s$ is the product of the primes $p\le\ell(Q)+\ell(R)$ and $\ell$ counts nonzero terms. Second, GHM's Theorem 4.1: if $F_n$ is the sequence obtained by attaching fixed pointed diagrams $(B,s)$ and $(C,t)$ to the two ends of $A_n$ (with $F_n(x)\in\mathbb{Z}[x]$ for all $n$), then either every $F_n$ is spherical or affine (in that case, if connected, the family is a re-indexing of one of the standard spherical or affine series $A_n,B_n,D_n,\widetilde{B}_n,\widetilde{C}_n,\widetilde{D}_n$), or the cyclotomic factor of $F_n(x)$ is periodic in $n$ for all $n\gg 0$. The proof rests on the two-term identity $F_n(x)(x-1)=x^{n+1}Q(x)-R(x)$ with $Q(x)=(B(x)-\widehat{B}(x))(C(x)-\widehat{C}(x))$ and $R(x)=(x\widehat{B}(x)-B(x))(x\widehat{C}(x)-C(x))$ fixed polynomials built from the decorations, together with a spectral monotonicity result under path extension due to Hoffman and Smith (their 1975 result is the source's Proposition 4.5). For the family $E_n$ itself, the identity reads $E_n(x)(x-1)=x^{n-2}(x^3-x-1)+(x^3+x^2-1)$, and GHM combined the bound $k\le 180$, separability of $E_n(x)$ for $n\ne 9$, and a finite check for $n\le 182$ to obtain, for all $n\ne 9$: the cyclotomic factor $C_n$ is the least common multiple of $\Phi_2$, $\Phi_3$ and the spherical polynomials $E_i$ ($3\le i\le 8$) that divide $E_n(x)$; $E_i\mid E_n$ iff $n\equiv i\pmod{h_i}$, where $h_i$ is the Coxeter number, that is, the order of the Coxeter element of $E_i$; and $\Phi_2\mid E_n$ iff $n$ is odd, $\Phi_3\mid E_n$ iff $3\mid n$. Consequently $C_n$ depends only on $n\bmod 360$, and in the authors' words these roots of unity come exclusively from splittings of $E_n$ into spherical subdiagrams. (Here $E_n$ is spherical for $3\le n\le 8$, affine for $n=9$ - the only $E_n$ whose Coxeter polynomial has multiple roots - and hyperbolic for $n\ge 10$.) What is not known is which general decoration data $((B,s),(C,t))$ force the same phenomenon, and how to make the eventual periodicity of Theorem 4.1 effective: the paper states that the dichotomy result, made effective, is what would reduce computations like Theorem 1.1 to a finite computation.

Problem Statement

Let $F_n$ be the sequence of Coxeter diagrams obtained by attaching two fixed pointed Coxeter diagrams $(B,s)$ and $(C,t)$ to the two ends of the path $A_n$, with $F_n(x)\in\mathbb{Z}[x]$ for all $n$ (the setting of Theorem 4.1 of Gross-Hironaka-McMullen; the source describes the same class of examples as sequences of Coxeter diagrams where nodes are added to a separating edge). As in the background, $\widehat{B}(x)$ and $\widehat{C}(x)$ denote the Coxeter polynomials of the diagrams obtained by deleting the basepoints $s$ and $t$. The common objective is a complete, structural and computable description of the cyclotomic part of the Coxeter polynomials $F_n(x)$, generalizing the description Gross-Hironaka-McMullen obtained for the $E_n$ family. Two linked deliverables are requested.

(1) Spherical-origin criterion. Following the source's usage ('the cyclotomic factors ... come exclusively from its spherical subdiagrams'; the mechanism is its Theorem 1.5 on joins), say that the family has the spherical-origin property if the following holds for every member $F_n$ that is neither spherical nor affine: whenever a root of unity $\zeta$ is a zero of $F_n(x)$, there is a vertex $v$ of $F_n$ such that (a) deleting $v$ separates $F_n$ into connected components $G_1,\dots,G_m$ with $m\ge 2$, the vertex $v$ being adjacent to exactly one vertex of each component (so that $F_n$ is the join of the pointed diagrams obtained from the components $G_i$ by re-attaching $v$), and (b) $\zeta$ is a common zero of the Coxeter polynomials of two or more of the components $G_i$, each of which is spherical (finite Coxeter group, positive-definite form). The spherical and affine members are exempted because their Coxeter polynomials are entirely cyclotomic for trivial reasons that need not be mediated by joins; in the source's family, $E_n$ is spherical for $3\le n\le 8$, affine for $n=9$, and hyperbolic for $n\ge 10$. Find a general condition - combinatorial, algebraic, or in terms of the decoration data $((B,s),(C,t))$ or the associated polynomials $Q(x)=(B(x)-\widehat{B}(x))(C(x)-\widehat{C}(x))$ and $R(x)=(x\widehat{B}(x)-B(x))(x\widehat{C}(x)-C(x))$ of the two-term identity $F_n(x)(x-1)=x^{n+1}Q(x)-R(x)$ - which guarantees that the spherical-origin property holds for the whole family. The condition must be correct (with proof), must be satisfied by the $E_n$ decorations and, combined with the spherical vertex-splittings it identifies, must reproduce the source's Theorem 1.1 for all $n\ne 9$ (cyclotomic orders exactly $\{2,3,5,8,12,18,30\}$, dependence only on $n\bmod 360$), and must be non-vacuous (it must hold for an infinite family of hyperbolic diagrams beyond $E_n$).

(2) Effective periodicity. Make the dichotomy theorem effective for these families: given any decoration $((B,s),(C,t))$ with $F_n(x)\in\mathbb{Z}[x]$, provide an explicit, terminating procedure that outputs either (a) a certificate that every $F_n$ is spherical or affine (by the source's Theorem 1.7/4.1 such a family is a re-indexing of one of the standard spherical or affine series $A_n,B_n,D_n,\widetilde{B}_n,\widetilde{C}_n,\widetilde{D}_n$), or (b) explicit, computably bounded integers $N$ and $M$, in terms of the decoration data, such that the cyclotomic factor of $F_n(x)$ depends only on $n\bmod M$ for all $n\ge N$, together with a terminating finite verification procedure that certifies this modulus for the given family. The two deliverables are jointly aimed at replacing the case-by-case Mann-type analyses currently needed for each new family by a general, algorithmic theory of the cyclotomic spectrum of decorated $A_n$ families.

References

1. Benedict H. Gross, Eriko Hironaka, Curtis T. McMullen, Cyclotomic factors of Coxeter polynomials, Journal of Number Theory 129 (2009), no. 5, 1034-1043. DOI: 10.1016/j.jnt.2008.09.021, https://doi.org/10.1016/j.jnt.2008.09.021 (author-hosted full text: https://people.math.harvard.edu/~ctm/home/text/papers/en/en.pdf)
2. Henry B. Mann, On linear relations between roots of unity, Mathematika 12 (1965), no. 2, 107-117. DOI: 10.1112/s0025579300005210, https://doi.org/10.1112/s0025579300005210
3. Norbert A'Campo, Sur les valeurs propres de la transformation de Coxeter, Inventiones Mathematicae 33 (1976), no. 1, 61-67. DOI: 10.1007/bf01425505, https://doi.org/10.1007/bf01425505
4. Eric Bedford, Kyounghee Kim, Dynamics of rational surface automorphisms: linear fractional recurrences, Journal of Geometric Analysis 19 (2009), no. 3, 553-583. DOI: 10.1007/s12220-009-9077-8, https://doi.org/10.1007/s12220-009-9077-8 (arXiv preprint math/0611297, https://arxiv.org/abs/math/0611297)
5. Frank Calegari, Scott Morrison, Noah Snyder, Cyclotomic integers, fusion categories, and subfactors, Communications in Mathematical Physics 303 (2011), no. 3, 845-896. DOI: 10.1007/s00220-010-1136-2, https://doi.org/10.1007/s00220-010-1136-2 (arXiv:1004.0665, https://arxiv.org/abs/1004.0665)
6. David Penneys, James E. Tener, Subfactors of index less than 5, part 4: vines, International Journal of Mathematics 23 (2012), no. 3, 1250017. DOI: 10.1142/S0129167X11007641, https://doi.org/10.1142/S0129167X11007641 (arXiv:1010.3797, https://arxiv.org/abs/1010.3797)
7. Charalampos A. Evripidou, Coxeter polynomials of Salem trees, Colloquium Mathematicum 141 (2015), no. 2, 209-226. DOI: 10.4064/cm141-2-6, https://doi.org/10.4064/cm141-2-6 (arXiv:1401.3128, https://arxiv.org/abs/1401.3128)
8. Chris Smyth, Seventy years of Salem numbers, Bulletin of the London Mathematical Society 47 (2015), no. 3, 379-395. DOI: 10.1112/blms/bdv027, https://doi.org/10.1112/blms/bdv027 (arXiv:1408.0195, https://arxiv.org/abs/1408.0195; the arXiv version is titled 'Survey article: Seventy years of Salem numbers', the same work)
9. Horst Brunotte, Jorg M. Thuswaldner, Salem numbers from a class of star-like trees, Journal of Number Theory 153 (2015), 85-96. DOI: 10.1016/j.jnt.2015.01.005, https://doi.org/10.1016/j.jnt.2015.01.005 (arXiv:1501.05557, https://arxiv.org/abs/1501.05557)
10. Jeffrey Diller, Kyounghee Kim, Entropy of real rational surface automorphisms, Experimental Mathematics 30 (2021), no. 2, 172-190. DOI: 10.1080/10586458.2018.1516581, https://doi.org/10.1080/10586458.2018.1516581 (arXiv:1710.07665, https://arxiv.org/abs/1710.07665)
11. Kyounghee Kim, The dynamical degrees of rational surface automorphisms, Conformal Geometry and Dynamics 28 (2024), 62-87. DOI: 10.1090/ecgd/392, https://doi.org/10.1090/ecgd/392 (arXiv:2211.09662, https://arxiv.org/abs/2211.09662)
12. Saieed Akbari, Keivan Mallahi-Karai, An infinite family of trees with irreducible characteristic polynomials, preprint (2026). arXiv:2607.23162, https://arxiv.org/abs/2607.23162
13. A. J. Hoffman, J. H. Smith, On the spectral radii of topologically equivalent graphs, in: Recent Advances in Graph Theory (Proc. Second Czechoslovak Symposium, Prague, 1974), Academia, Prague, 1975, pp. 273-281. Bibliographic data as given in the reference list of Gross-Hironaka-McMullen (DOI: 10.1016/j.jnt.2008.09.021), whose Proposition 4.5 uses [HS, Lem. 2.3 and Prop. 2.4]