## 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
Full Text:

### Online Encyclopedia of Integer Sequences:

Coxeter number for the reflection group E_n.

### References:

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