An algebraic integration for Mahler measure

Abstract

There are many examples of several-variable polynomials whose Mahler measure is expressed in terms of special values of polylogarithms. These examples are expected to be related to computations of regulators, as observed by Deninger [D], and later Rodriguez-Villegas [R-V], and Maillot [M]. While Rodriguez-Villegas made this relationship explicit for the two variable case, it is our goal to understand the three variable case and shed some light on the examples with more variables.Classification Primary 11G55; Secondary 19F99.

Zeta Mahler measures

“…It is known that the Mahler measure has interesting connections with zeta/L values, (multiple) polylogarithms and multiple sine functions, see [B,D,L1,L2,O,RV,S,V] and references therein.…”

Mahler measure and elliptic curve L-functions at s = 3

“…In our cases we will have that m.P / D 0 and this term will be ignored. (See [12] for more details. )…”

Mahler measures and computations with regulators

“…In [Lal07] we developed these ideas for some three-variable cases involving evaluations of trilogarithms and shed some light on how these computations could be carried on for more variables. Let us consider the three-variable case.…”

“…In this work we explore the techniques of [Lal07] in application to several concrete examples. Most of the examples that we study here are in three variables, but we also include an example in four variables, and we apply the method to compute some generalized Mahler measures.…”

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In this work we apply the techniques that were developed in [Lal07] in order to study several examples of multivariable polynomials whose Mahler measure is expressed in terms of special values of the Riemann zeta function or Dirichlet L-series. The examples may be understood in terms of evaluations of regulators.

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