Contact: Pieter Moree (moree@mpim-bonn.mpg.de)

The Mahler measure of a multivariate Laurent polynomial P is a real number which measures the arithmetic complexity of P, and appears in many areas of mathematics, ranging from the Iwasawa theory of knots to ergodic theory. The set of real numbers which can be expressed as the Mahler measure of a polynomial with integer coefficients has some very interestsing topological properties, as observed by Boyd. In particular, one expects it to be closed. In this talk, based on joint work with François Brunault, Antonin Guilloux and Mahya Mehrabdollahei, I will show how one can produce many interesting Cauchy sequences of Mahler measures, which converge to a Mahler measure, therefore respecting Boyd's conjecture. We are able moreover to give an explicit bound for the error term in the convergence of these sequences, and a full asymptotic exapansion for one explicit family of polynomials, whose Mahler measure can be expressed in terms of the Bloch-Wigner dilogarithm evaluated at certain roots of unity. This is due to the fact that these polynomials share the property of being exact, which was introduced in the work of Maillot and Lalin. In the second part of my talk, based on work in progress with François Brunault, I will introduce this notion briefly, and a generalization of it, and their utilities in predicting links between Mahler measures and special values of L-functions.

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