Mahler measure of Pd polynomials

This article investigates the Mahler measure of a family of 2variate polynomials, denoted by P-d, for d > 1, unbounded in both degree and genus. By using a closed formula for the Mahler measure [13], we are able to compute m(P-d), for arbitrary d, as a sum of the values of dilogarithm at special roots of unity. We prove that m(P-d) converges, and the limit is proportional to (3), where is the Riemann zeta function. The proof we give is computational and based on the estimation of the error of Riemann sums of a bivariate function. We describe a second possible shorter proof based on a conjectural generalization of the theorem of Boyd-Lawton and a result of D'Andrea and Lalin [11].