Evaluations of the areal Mahler measure of multivariable polynomials

TL;DR

This paper explores the areal Mahler measure of multivariable polynomials, providing new evaluations that relate to special values of L-functions by integrating over product of disks instead of the torus.

Contribution

It introduces novel evaluations of the areal Mahler measure for multivariable polynomials, extending classical Mahler measure concepts to new integral domains.

Findings

Identified specific polynomials with explicit Mahler measure evaluations

Connected Mahler measure values to special L-function values

Extended Mahler measure analysis from the torus to product of disks

Abstract

We exhibit some nontrivial evaluations of the areal Mahler measure of multivariable polynomials, defined by Pritsker [Pri08] by considering the integral over the product of unit disks instead of the unit torus as in the standard case. As in the case of the classical Mahler measure, we find examples yielding special values of $L$-functions.