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,xn) 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.
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