On the Mahler measure of hyperelliptic families

Abstract

We prove Boyd's "unexpected coincidence" of the Mahler measures for two families of two-variate polynomials defining curves of genus 2. We further equate the same measures to the Mahler measures of polynomials $y^3-y+x^3-x+kxy$ whose zero loci define elliptic curves for $k\ne0,\pm3$.