Igusa zeta functions and Mahler measure

Abstract

Igusa zeta functions are a type of generating function that counts the number of solutions to polynomial equations, which can be written as an integral over the p-adic numbers. They were proven to be rational in 1974 by Igusa, a fact which since has been proven twice more using other sophisticated techniques. Mahler measure is a well-known invariant associated to Laurent polynomials.

In my paper I prove some results about Igusa zeta functions, including a one-dimensional approximation theorem, discuss some intriguing connections to Mahler measure and then prove some results, which have been conjectured for Mahler measure, in the Igusa zeta function setting.