Polynomials with small elliptic Mahler measure via genetic algorithms

Abstract

For a minimal polynomial $f$ we denote by $M(f)$ the Mahler measure of the roots of $f$. The classical Lehmer conjecture is concerned with finding a definitive lower bound for $M(f)$. Lehmer's polynomial is known to have the lowest Mahler measure for all polynomials. We look at a version of Lehmer's conjecture involving elliptic curves and investigate the corresponding Mahler measures, looking for those polynomials with minimal Mahler measure on various elliptic curves. We detail the results we have found using genetic algorithms.