Finding new small degree polynomials with small Mahler measure by genetic algorithms

Abstract

In this paper, we propose a new application of genetic-type algorithms to find monic, irreducible, non-cyclotomic integer polynomials with \textit {small degree} and Mahler measure less than $1.3$, which do not appear in Mossinghoff's list of all known polynomials with degree at most 180 and Mahler measure less than 1.3 {Mossinghoff}. The primary focus lies in finding such polynomials of small degree. In particular, the list referred to above is known to be complete through degree 44, and we show that it is not complete from degree 46 on by supplying two new polynomials of small Mahler measure, of degrees 46 and 56. We also provide a large list of polynomials of small Mahler measure of degrees up to 180 which, although discovered by us through the use of a method described in Boyd and Mossinghoff {Boyd and Mossinghoff} based on limit points of small Mahler measures, do not appear on Mossinghoff's list \cite {Mossinghoff 07}. Finally, we verify that our new polynomials of degrees 46 and 56 cannot be produced from the known small limit points.