Generalized Mahler measures of Laurent polynomials

TL;DR

This paper extends the concept of Mahler measures to Laurent polynomials in multiple variables, establishing relations with standard measures, and explicitly computes measures for a specific polynomial family using special functions.

Contribution

It introduces a generalized Mahler measure for Laurent polynomials that do not vanish on the torus and computes explicit measures for a family of polynomials, extending previous work.

Findings

Derived relations between standard and generalized Mahler measures.

Explicitly calculated the generalized Mahler measure for $Q_4$ using the Bloch-Wigner dilogarithm.

Extended the measure concepts to multivariable polynomials.

Abstract

Following the work of Lal\'in and Mittal on the Mahler measure over arbitrary tori, we investigate the definition of the generalized Mahler measure for all Laurent polynomials in two variables when they do not vanish on the integration torus. We establish certain relations between the standard Mahler measure and the generalized Mahler measure of such polynomials. Later we focus our investigation on a tempered family of polynomials originally studied by Boyd, namely $Q_{r}(x, y) = x + \frac{1}{x} + y + \frac{1}{y} + r$ with $r \in \mathbb{C},$ and apply our results to this family. For the $r = 4$ case, we explicitly calculate the generalized Mahler measure of $Q_4$ over any arbitrary torus in terms of special values of the Bloch-Wigner dilogarithm. Finally, we extend our results to the several variable setting.