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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 \[ F_n(x)=\frac{x^{n+1}Q(x)-R(x)}{x-1}. \]

MSC:

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.

References:

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