Lehmer’s problem for polynomials with odd coefficients | Annals of Mathematics

Abstract We prove that if f(x)=∑n−1k=0akxk is a polynomial with no cyclotomic factors whose coefficients satisfy ak≡1 mod 2 for 0≤k1+log32n, 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}.

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数学年刊(Annals of Mathematics)

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