Asymptotic expansion of the difference of two Mahler measures

TL;DR

This paper proves that for most polynomials, the difference in Mahler measures between P(x,y) and P(x,x^n) admits a unique asymptotic expansion in powers of 1/n, with coefficients involving polylogarithms of algebraic numbers.

Contribution

It generalizes Boyd’s result by establishing the existence, uniqueness, and explicit formula for the asymptotic expansion of Mahler measure differences.

Findings

The expansion is unique for almost all polynomials.

Coefficients are linear combinations of polylogarithms of algebraic numbers.

The expansion generalizes previous results on Mahler measures.

Abstract

We show that for almost every polynomial P(x,y) with complex coefficients, the difference of the logarithmic Mahler measures of P(x,y) and P(x,x^n) can be expanded in a type of formal series similar to an asymptotic power series expansion in powers of 1/n. This generalizes a result of Boyd. We also show that such an expansion is unique and provide a formula for its coefficients. When P has algebraic coefficients, the coefficients in the expansion are linear combinations of polylogarithms of algebraic numbers, with algebraic coefficients.