Asymptotic expansion of the difference of two 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.
 Publication:

arXiv eprints
 Pub Date:
 November 2011
 arXiv:
 arXiv:1111.0241
 Bibcode:
 2011arXiv1111.0241C
 Keywords:

 Mathematics  Number Theory;
 11C08 (Primary) 11R06;
 41A60 (Secondary)
 EPrint:
 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