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An algebraic integration for Mahler measure. (English) Zbl 1196.11096

Building on some earlier work of Deninger, Rodriguez-Villegas and Maillot the author developed some algebraic machinery (based on calculation of some differential forms) which is useful in proving some explicit formulas for the Mahler measures of polynomials in several variables. In this paper, the author concentrates on the three variable case. As an example, she recovers the logarithmic Mahler measure of the rational function \(z-(1-x)^m(1-y)^n(1-xy)^{-mn}.\) In the four dimensional case, the result is applied to the polynomial \((1-x)(1-y)-(1-w)(1-z)\) whose logarithmic Mahler measure is known to be \(9\zeta(3)/2\pi^2.\) In the \(n\)-dimensional case she gives a general conjecture. This conjecture if proved would explain the nature of some \(n\)-variable examples. She then works with the family of \((n+1)\)-variable rational functions \(z=\prod_{j=1}^n (1-x_j)/(1+x_j)\).

MSC:

11G55 Polylogarithms and relations with \(K\)-theory
19F99 \(K\)-theory in number theory
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References:

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