Limits of Mahler measures in multiple variables

TL;DR

This paper proves convergence properties of Mahler measures for sequences of Laurent polynomials obtained through monomial substitutions, extending prior results and providing explicit error bounds and asymptotic expansions.

Contribution

It generalizes previous work by Boyd and Lawton on Mahler measure convergence to multivariate cases and offers explicit error bounds and asymptotic formulas.

Findings

Sequences of Mahler measures converge to the Mahler measure of the original polynomial.

An explicit upper bound for the convergence error is established.

A full asymptotic expansion is derived for a family of 2-variable polynomials.

Abstract

We prove that certain sequences of Laurent polynomials, obtained from a fixed Laurent polynomial P by monomial substitutions, give rise to sequences of Mahler measures which converge to the Mahler measure of P. This generalizes previous work of Boyd and Lawton, who considered univariate monomial substitutions. We provide moreover an explicit upper bound for the error term in this convergence, generalizing work of Dimitrov and Habegger, and a full asymptotic expansion for a family of 2-variable polynomials, whose Mahler measures were studied independently by the third author.