Lehmer’s problem for polynomials with odd coefficients

Abstract

We prove that if $f(x)=\sum_{k=0}^{n-1} a_k x^k$ is a polynomial with no cyclotomic factors whose coefficients satisfy $a_k\equiv1$ mod 2 for $0\leq k\lt n$, then Mahler’s measure of $f$ satisfies \[ \log {\rm M}(f) \geq \frac{\log 5}{4}\left(1-\frac{1}{n}\right). \] This resolves a problem of D. H. Lehmer [12] for the class of polynomials with odd coefficients. We also prove that if $f$ has odd coefficients, degree $n-1$, and at least one noncyclotomic factor, then at least one root $\alpha$ of $f$ satisfies \[ \left\lvert\alpha\right\rvert > 1 + \frac{\log3}{2n}, \] resolving a conjecture of Schinzel and Zassenhaus [21] for this class of polynomials. More generally, we solve the problems of Lehmer and Schinzel and Zassenhaus for the class of polynomials where each coefficient satisfies $a_k\equiv1$ mod $m$ for a fixed integer $m\geq2$. We also characterize the polynomials that appear as the noncyclotomic part of a polynomial whose coefficients satisfy $a_k\equiv1$ mod $p$ for each $k$, for a fixed prime $p$. Last, we prove that the smallest Pisot number whose minimal polynomial has odd coefficients is a limit point, from both sides, of Salem [19] numbers whose minimal polynomials have coefficients in $\{-1,1\}$.