Mahler measure of $P_d$ polynomials

Abstract

This article investigates the Mahler measure of a family of 2-variate polynomials, denoted by $P_d,d\ge 1$, unbounded in both degree and genus. By using a closed formula for the Mahler measure introduced in "Volume function and Mahler measure of exact polynomials" (by Guilloux and Marché), we are able to compute $m(P_d)$, for arbitrary $d$, as a sum of the values of dilogarithm at special roots of unity. We prove that $m(P_d)$ converges and the limit is proportional to $\zeta (3)$, where $\zeta$ is the Riemann zeta function.