Lehmer’s problem for polynomials with odd coefficients


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 $\{-1,1\}$.


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