Mahler measure of $P_d$ polynomials

TL;DR

This paper computes the Mahler measure of a family of bivariate polynomials, showing that it converges to a value proportional to the Riemann zeta function at 3, using a closed-form formula involving dilogarithms.

Contribution

It provides a closed-form computation of Mahler measures for the polynomials $P_d$, extending understanding of their asymptotic behavior and connection to special values of the dilogarithm and zeta function.

Findings

Mahler measure $m(P_d)$ converges as $d$ increases

Limit of $m(P_d)$ is proportional to $zeta(3)$

Explicit formula involves dilogarithm at roots of unity

Abstract

This article investigates the Mahler measure of a family of 2-variate polynomials, denoted by $P_d, d\geq 1$, 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 March\'e), 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.