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# Mahler's Measure and the Dilogarithm (I)

Published:2002-06-01
Printed: Jun 2002
• David W. Boyd
• Fernando Rodriguez-Villegas
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## Abstract

An explicit formula is derived for the logarithmic Mahler measure $m(P)$ of $P(x,y) = p(x)y - q(x)$, where $p(x)$ and $q(x)$ are cyclotomic. This is used to find many examples of such polynomials for which $m(P)$ is rationally related to the Dedekind zeta value $\zeta_F (2)$ for certain quadratic and quartic fields.
 MSC Classifications: 11G40 - $L$-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture [See also 14G10] 11R06 - PV-numbers and generalizations; other special algebraic numbers; Mahler measure 11Y35 - Analytic computations