Some relations of Mahler measure with hyperbolic volumes and special values of L-functions



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We 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.