A generalization of a theorem of Boyd and Lawton

TL;DR

This paper extends Boyd and Lawton's theorem, showing that various generalized Mahler measures of multivariate polynomials can be approximated by univariate cases, broadening the understanding of Mahler measure limits.

Contribution

It proves the analogous limit results for generalized, multiple, and higher Mahler measures, expanding the classical theorem to new types of Mahler measures.

Findings

Generalized Mahler measure limit established

Multiple Mahler measure approximation proven

Higher Mahler measure convergence demonstrated

Abstract

The Mahler measure of a nonzero $n$-variable polynomial $P$ is the integral of $\log|P|$ on the unit $n$-torus. A result of Boyd and Lawton says that the Mahler measure of a multivariate polynomial is the limit of Mahler measures of univariate polynomials. We prove the analogous result for different extensions of Mahler measure such as generalized Mahler measure (integrating the maximum of $\log|P|$ for possibly different $P$'s), multiple Mahler measure (involving products of $\log|P|$ for possibly different $P$'s), and higher Mahler measure (involving $\log^k|P|$).