Average Mahler’s measure andLpnorms of unimodular polynomials

Abstract

A polynomial f 2 CTzU is unimodular if all its coefficients have unit modulus. Let Un denote the set of unimodular polynomials of degree n 1, and let U n denote the subset of reciprocal unimodular polynomials, which have the property that f.z/D !z n 1 f.1=z/ for some complex number ! withj!jD 1. We study the geometric and arithmetic mean values of both the normalized Mahler’s measure M. f/= p n and L p normjj fjj p= p n over the sets Un and U n, and compute asymptotic values in each case. We show for example that both the geometric and arithmetic mean of the normalized Mahler’s measure approach e = 2 D 0:749306::: as n!1 for unimodular polynomials, and e = 2 = p 2D 0:529839::: for reciprocal unimodular polynomials. We also show that for large n, almost all polynomials in these sets have normalized Mahler’s measure or L p norm very close to the respective limiting mean value.