Limits of Mahler measures in multiple variables

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.

arXiv commentIn V2, the error term has been improved with respect to V1, with a new bound of independant interest given in Theorem 4.6. This bound replaces a reference to Dimitrov-Habegger in the previous version, and follows closely their method

References in corpus (11)

• Exponential growth of torsion in Abelian coverings
• Twisting $L^2$-invariants with finite-dimensional representations
• On the Mahler measure of Jones polynomials under twisting
• Higher Mahler measures and zeta functions
• Mahler measures and Fuglede--Kadison determinants
• On the Mahler measure of resultants in small dimensions
• Volume function and Mahler measure of exact polynomials
• On links with cyclotomic Jones polynomials
• A generalization of a theorem of Boyd and Lawton
• Lower bounds for Mahler measure that depend on the number of monomials
• Mahler measure of $P_d$ polynomials