Asymptotic expansion of the difference of two Mahler measures
Number Theory
2011-11-14 v2
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.
Keywords
Cite
@article{arxiv.1111.0241,
title = {Asymptotic expansion of the difference of two Mahler measures},
author = {John D. Condon},
journal= {arXiv preprint arXiv:1111.0241},
year = {2011}
}
Comments
25 pages. V2: Demoted previous Corollary 1 to a comment, after realizing that Boyd had already proved that bit. Made small corrections to Lemma 5, streamlined the proof of Lemma 9, and reworded section 9.3