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The Mahler measure of a family of polynomials of two variables. (Measure de Mahler d’une familie de polynômes de deux variables.) (French) Zbl 1111.11045

The logarithmic Mahler measure \(m(f)\) of a rational function of \(n\) variables is defined to be the average over the \(n\)-torus of \(\log |f| \). The author studies \(m(s):= m(P_s)\) for the family of polynomials \(P_s = (x^2 + x - 1)y^2 - sxy - x^2 + x + 1\). Although only real values of \(s\) are considered in the paper, the methods used extend to the case of complex \(s\) as well. She observes that \(m(s) = - \log \phi\), for \(0 < s \leq 2\), where \(\phi = (\sqrt 5 - 1)/2 \). She evaluates \(m(2\sqrt 5) = - \log \phi + (4/\pi)\text{Li}_2(\chi, \phi)\), where \(\text{Li}_2(\chi,z) = \sum_{n=1}^\infty \chi(n)z^n/n^2\), \( \chi(n)\) being the odd quadratic character of conductor \(4\). She observes that \(m'(s)\) is a residue in the sense of Poincaré and studies it by means of the Picard-Fuchs equation thus identifying it as the period of an elliptic curve. This leads to some integral formulas involving the complete elliptic integral.
Reference should be made to the paper of F. Rodriguez Villegas [Topics in number theory, Math. Appl., Dordr. 467, 17–48 (1999; Zbl 0980.11026)] which treats families of this type in more generality.

MSC:

11R06 PV-numbers and generalizations; other special algebraic numbers; Mahler measure
11G55 Polylogarithms and relations with \(K\)-theory

Citations:

Zbl 0980.11026
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