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 $a_{k}$ ≡ 1 mod 2 for 0 ≤ k &lt n, then Mahler's measure of f satisfies $\text{log M}(f)\geq \frac{\text{log}5}{4}(1-\frac{1}{n})$ . 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 α of f satisfies $|\alpha |1+\frac{\text{log}3}{2n}$ , 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 $a_{k}\equiv 1$ mod m for a fixed integer m ≥ 2. We also characterize the polynomials that appear as the noncyclotomic part of a polynomial whose coefficients satisfy $a_{k}\equiv 1$ 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}.