On Lehmer's Problem

Abstract

Let $m\geq 2$ be an integer, and let $d\geq 2$ be an even integer. We study monic, reciprocal, non-cyclotomic polynomials $$ P(x)=x^d+a_{d-1}x^{d-1}+\cdots+a_1x+1\in\mathbb{Z}[x]. $$ For a fixed residue class $r\in\{0,1,\ldots,m-1\}$, we prove that there exists an effectively computable constant $c_{r,m}>1$ such that the Mahler measure $$ M(P)\geq c_{r,m} $$ for every irreducible polynomial $Q$ whose inner coefficients satisfy $$ a_j\equiv r\pmod m \qquad (1\leq j\leq d-1). $$ Moreover, we explicitly compute the constant $c_{r,m}$ in terms of $r$ and $m$. For the residue classes $r=2$, $r=3$, and $r=m-1$, we obtain improved lower bounds for suitable ranges of the modulus~$m$. We also consider monic, reciprocal, irreducible, non-cyclotomic polynomials whose inner coefficients belong to two distinct residue classes modulo $m$. More precisely, if the inner coefficients with odd indices are congruent to $a\pmod m$ and those with even indices are congruent to $b\pmod m$, where $a\not\equiv b\pmod m$, then we prove the existence of a constant $c_{m,a,b}>1$ such that the Mahler measure $$ M(P)\geq c_{m,a,b}. $$ These results solve Lehmer's problem for several families of polynomials. We also construct infinitely many examples in every even degree $d\geq 2$ of monic, reciprocal, irreducible, non-cyclotomic polynomials over $\mathbb{Q}$, together with explicit information about their Mahler measures. Our proofs rely on Y. Bilu's equidistribution theorem and its effective version proved by C. D'Andrea \emph{et al.}, as well as on a method developed by P. Borwein \emph{et al.} While it is natural to consider Bilu's equidistribution results in conjunction with the Borwein {\it et al}. method to Lehmer's problem, we are not aware of any previous work in the literature that employs this particular approach.