Séminaire "Equations différentielles"
The Mahler measure of multivariate polynomials appears in several fields of mathematics including number theory. In particular there are deep conjectural links between Mahler measures of integer polynomials and special values of L-functions. The first theorem in this direction is due to Smyth (1981), who computed the Mahler measure of the polynomial 1+x+y in terms of the L-function of the Dirichlet character of conductor 3. This can be generalised to a class of 2-variable polynomials called exact polynomials. I will explain work in progress with Riccardo Pengo, where we introduce the degree of exactness of a polynomial P(x_1,...,x_n). Under certain conditions, the Mahler measure of P can then be interpreted as a Deligne period on a subvariety of the hypersurface P=0, whose codimension is equal to the exactness degree of P.
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Vasily Golyshev & Vladimir Roubtsov