Small Limit Points of Mahler's Measure

Abstract

Let $M(P(z_1,\dots,z_n))$ denote Mahler's measure of the polynomial $P(z_1,\dots,z_n)$. Measures of polynomials in $n$ variables arise naturally as limiting values of measures of polynomials in fewer variables. We describe several methods for searching for polynomials in two variables with integer coefficients having small measure, demonstrate effective methods for computing these measures, and identify 48 polynomials $P(x,y)$ with integer coefficients, irreducible over $\Rats$, for which $1 < M(P(x,y)) < 1.37$.