Polynomials with small Mahler measure

Abstract

We describe several searches for polynomials with integer coefficients and small Mahler measure. We describe the algorithm used to test Mahler measures. We determine all polynomials with degree at most 24 and Mahler measure less than 1.3 1.3 , test all reciprocal and antireciprocal polynomials with height 1 and degree at most 40, and check certain sparse polynomials with height 1 and degree as large as 181. We find a new limit point of Mahler measures near 1.309 1.309 , four new Salem numbers less than 1.3 1.3 , and many new polynomials with small Mahler measure. None has measure smaller than that of Lehmer’s degree 10 polynomial.