Lehmer’s problem for polynomials with odd coefficients
Peter Borwein, Edward Dobrowolski, Michael J. Mossinghoff · Annals of Mathematics · 2007
We prove that if f (x) = n-1 k=0 a k x k is a polynomial with no cyclotomic factors whose coefficients satisfy a k ≡ 1 mod 2 for 0 ≤ k 1 + 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 ≡ 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 ≡ 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}.*The first author was supported in part by NSERC of Canada and MITACS.The authors thank the Banff International Research Station for hosting the workshop on "The many aspects of Mahler's measure