Some remarks on the Lehmer conjecture

Abstract

In 1933, Lehmer exhibited the polynomial

$$\begin{aligned} L(z)=z^{10} + z^9 - z^7 - z^6 - z^5 - z^4 - z^3 + z + 1 \end{aligned}$$

with Mahler measure $\mu _0>1$. Then he asked if $\mu _0$ is the smallest Mahler measure, not 1. This question became known as the Lehmer conjecture and it was apparently solved in the positive, while this paper was in preparation [19]. In this paper we consider those polynomials of the form $\chi _A$, that is, Coxeter polynomials of a finite dimensional algebra A (for instance $L(z)=\chi _{\mathbb {E}_{10}}$). A polynomial in $\mathbb {Z}[T]$ which is either cyclotomic or with Mahler measure $\ge \mu _0$ is called a Lehmer polynomial. We give some necessary conditions for a polynomial to be Lehmer. We show that A being a tree algebra is a sufficient condition for $\chi _A$ to be Lehmer.