Lehmer's problem for polynomials with odd coefficients

Abstract

We prove that if f(x) = Σk=0n-1 a kxk is a polynomial with no cyclotomic factors whose coefficients satisfy ak = 1 mod 2 for O ≤ k 1 + 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 ak = 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 ak = 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}.