Browsing by Subject "Logarithmic functions"
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Item Mahler measure evaluations in terms of polylogarithms(2004) Condon, John Donald; Rodríguez Villegas, FernandoWe 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.Item Some relations of Mahler measure with hyperbolic volumes and special values of L-functions(2005) Lalín, Matilde Noemí; Rodriguez-Villegas, FernandoWe construct families of polynomials of up to five variables whose Mahler measures are given in terms of multiple polylogarithms. The formulas are homogeneous and their weight coincides with the number of variables of the corresponding polynomial. Next, we fix the coefficients of these families and find some n-variable polynomial families whose Mahler measure is expressed in terms of polylogarithms, zeta functions, and Dirichlet L-functions. We also develop examples of formulas where the Mahler measure of certain polynomial may be interpreted as the volume of a hyperbolic object. The examples involving polylogarithms, zeta functions and Dirichlet L-functions are expected to be related to computations of regulators in motivic cohomology as observed by Deninger, and later Rodriguez-Villegas and Maillot. While RodriguezVillegas made this relationship explicit for the two variable case, we have described in detail the three variable case and we expect to extend our ideas to several variables.