A Generalization of a Theorem of Boyd and Lawton

Abstract

Abstract.The Mahler measure of a nonzero n-variable polynomial P is the integral of log |P| on the unit n-torus. A result of Boyd and Lawton says that the Mahler measure of a multivariate polynomial is the limit of Mahler measures of univariate polynomials. We prove the analogous result for different extensions of Mahler measure such as generalized Mahler measure (integrating the maximum of log |P| for possibly different P’s), multiple Mahler measure (involving products of log |P| for possibly different P’s), and higher Mahler measure (involving logk |P|).