Riccardo Pengo

Limits of Mahler measures and (successively) exact polynomials

Mahler's measure is a height function of fundamental importance in Diophantine geometry, protagonist of a celebrated problem posed by Lehmer. The work of Boyd has shown that Lehmer's problem can be approached by studying Mahler measures of multivariate polynomials, and that the latter are often linked to special values of L-functions. In this seminar, I will talk about a generalization of the work of Boyd, obtained jointly with François Brunault, Antonin Guilloux and Mahya Mehrabdollahei, in which we find a class of sequences of polynomials whose Mahler measures converge. Furthermore, we provide an explicit upper bound for the error term, and an asymptotic expansion for a particular family of polynomials, whose terms share all the peculiar property of being "exact". If time permits, I will explain more in detail this notion of exactness, and talk about a generalization of it (the notion of "successive exactness"), which I studied jointly with François Brunault.