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. For a video summary of this paper, please click here or visit http://www.youtube.com/watch?v=8VB7gxgPqdc .