The Mahler Measure of Parametrizable Polynomials

Abstract

Our aim is to explain instances in which the value of the logarithmic Mahler measure of a polynomial can be written in an unexpectedly neat manner. To this end we examine polynomials defining rational curves, which allows their zero-locus to be parametrized. In these instances the unusual simplification is a consequence of the Galois descent property for Bloch groups. This principle enables one to explain why the arguments of the dilogarithm function depend only on the points where the rational curve intersects the torus |x|=|y|=1. In the process we also present a general method for computing the Mahler measure of any parametrizable polynomial.