Mahler measure evaluations in terms of polylogarithms

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2004

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Condon, John Donald

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We prove a conjecture of Boyd by showing that the logarithmic Mahler measure of a certain integer polynomial in three variables is equal to 28 ζ(3). 5π2 The proof proceeds by expressing the Mahler measure as a combination of integrals of one-variable logarithmic forms, evaluating these in terms of poly- logarithm functions at algebraic arguments, and using identities to simplify the expression. Next, we indicate how the techniques used in the previous example can be applied to give Mahler measure evaluations in terms of polylogarithms for two families of three variable polynomials. As an example, we work out the details for a four-parameter subfamily. Finally, we discuss an alternative, more algebraic approach to this sort of calculation. This method, developed by Rodriguez Villegas, Boyd, Maillot and others, relies on showing that certain elements of algebraic K-groups are equal to zero. We reinterpret our original problem in this context and consider attempts at its resolution.

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