Cyclotomic factors of Coxeter polynomials. (English) Zbl 1183.11065

For \(n\geq 3\) the Coxeter polynomial of a certain diagram \(E_n\) can be defined as \[ E_n(x)=\frac{x^{n-2}(x^3-x-1)+x^3+x^2-1}{x-1}. \] This polynomial has at most one root outside the unit circle, so it can be written uniquely as a product \(E_n (x) = C_n (x)S_n (x)\) with the cyclotomic factor \(C_n (x) =\prod_i \Phi_i (x)\) and the Salem factor \(S_n (x)\). For instance, \(E_7(x) = \Phi_2 (x)\Phi_{18}(x)\), \(E_8(x) = \Phi_{30}(x)\), whereas for \(n = 10\) we have \[ E_{10} (x) = S_{10}(x) = 1 + x - x^3 - x^4 - x^5 - x^6 - x^7 + x^9 + x^{10} \] which determines the smallest known Salem number \(1.17628\dots\). The authors prove that for \(n \neq 9\) the cyclotomic factor \(C_n(x)\) is the least common multiple of the polynomials \(\Phi_2(x), \Phi_3(x)\) and \(E_i(x)\), \(3\leq i\leq 8\), that divide \(E_n(x)\). They also prove that \(E_n(x)\) is divisible by \(\Phi_2(x)\) iff \(n\) is odd and is divisible by \(\Phi_3(x)\) iff \(3|n\) and determine all \(n\) for which \(E_n(x)\) is divisible by \(E_i (x)\), \(3\leq i\leq 8\). This implies that the cyclotomic factor \(C_n(x)\) only depends on \(n\) modulo 360 and that \(n - 15\leq \deg S_n\leq n\). For the proof, they first show that \(E_n(x)\) is separable for all \(n \neq 9\) and that \(E_n(\exp(2i/k)) = 0\) yields \(k\leq 180\). Finally, they generalize the results on \(E_n\) to some more general diagrams \(F_n \)with Coxeter polynomials \[ F_n(x)=\frac{x^{n+1}Q(x)-R(x)}{x-1}. \]


11R06 PV-numbers and generalizations; other special algebraic numbers; Mahler measure
12D05 Polynomials in real and complex fields: factorization
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Online Encyclopedia of Integer Sequences:

Coxeter number for the reflection group E_n.


[1] A’Campo, N., Sur les valeurs propres de la transformation de Coxeter, Invent. Math., 33, 61-67 (1976) · Zbl 0406.20041
[2] E. Bedford, K. Kim, Dynamics of rational surface automorphisms: Linear fractional recurrences, preprint, 2006; E. Bedford, K. Kim, Dynamics of rational surface automorphisms: Linear fractional recurrences, preprint, 2006 · Zbl 1185.37128
[3] Bourbaki, N., Groupes et algèbres de Lie, Ch. IV-VI (1981), Hermann: Masson · Zbl 0483.22001
[4] Cvetković, D.; Doob, M.; Sachs, H., Spectra of Graphs (1980), Academic Press · Zbl 0458.05042
[5] Cvetković, D.; Rowlinson, P., The largest eigenvalue of a graph: A survey, Linear Multilinear Algebra, 28, 3-33 (1990) · Zbl 0744.05031
[6] Hironaka, E., Salem-Boyd sequences and Hopf plumbing, Osaka J. Math., 43, 497-516 (2006) · Zbl 1109.57004
[7] Hironaka, E., Hyperbolic perturbations of algebraic links and small Mahler measure, (Singularities in Geometry and Topology 2004 (2007), Math. Soc. Japan), 77-94 · Zbl 1138.57003
[8] Hoffman, A. J.; Smith, J. H., On the spectral radii of topologically equivalent graphs, (Recent Advances in Graph Theory, Proc. Second Czechoslovak Sympos.. Recent Advances in Graph Theory, Proc. Second Czechoslovak Sympos., Prague, 1974 (1975), Academia), 273-281
[9] Humphreys, J. E., Reflection Groups and Coxeter Groups (1990), Cambridge Univ. Press · Zbl 0725.20028
[10] Mann, H. B., On linear relations between roots of unity, Mathematika, 12, 107-117 (1965) · Zbl 0138.03102
[11] McKee, J. F.; Rowlinson, P.; Smyth, C. J., Pisot numbers from stars, (Number Theory in Progress, vol. I (1999), de Gruyter), 309-319 · Zbl 0931.11041
[12] McKee, J. F.; Smyth, C. J., Pisot numbers, Mahler measure, and graphs, Experiment. Math., 14, 211-229 (2005) · Zbl 1082.11066
[13] McMullen, C., Coxeter groups, Salem numbers and the Hilbert metric, Publ. Math. Inst. Hautes Études Sci., 95, 151-183 (2002) · Zbl 1148.20305
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