Title: Special Functions and the Range of Mahler Measure

Chris Sinclair, Pacific Institute for the Mathematical Sciences

 

Abstract: The range of Mahler measure restricted to polynomials in C[x] and R[x] is studied by introducing two families of analytic functions. These analytic functions have meromorphic continuations to rational functions, and the values of these function allow us to compute the main term for estimates for the number of polynomials of fixed degree and bounded Mahler measure. Moreover the rational functions which appear can be represented as determinants (Pfaffians) of symmetric (antisymmetric) Gram matrices with respect to an inner product (skew inner product) associated to Mahler measure. The identities which arise are related to certain quantities of interest in the theory of special functions and random matrix theory.