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 $\{-1,1\}$.

Authors

Peter Borwein

Department of Mathematics
Simon Fraser University
Burnaby BC V5A 1S6
Canada

Edward Dobrowolski

Department of Mathematics
College of New Caledonia
Prince George, B.C. V2N 1P8
Canada

Michael J. Mossinghoff

Department of Mathematics
Davidson College
Davidson, NC 28035
United States