Average Mahler’s measure and L_p norms of unimodular polynomials

Kwok-Kwong Stephen Choi and Michael J. Mossinghoff

Vol. 252 (2011), No. 1, 31–50

DOI: 10.2140/pjm.2011.252.31

Abstract

A polynomial f ∈ ℂ[z] is unimodular if all its coefficients have unit modulus. Let U_n 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) = ωzⁿ⁻¹f(1∕z) for some complex number ω with |ω| = 1. We study the geometric and arithmetic mean values of both the normalized Mahler’s measure M(f)∕ and L _p norm ||f||_p∕ over the sets U_n 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^−γ∕2 = 0.749306… as n →∞ for unimodular polynomials, and e^−γ∕2∕ = 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.

Keywords

mean Mahler’s measure, mean L_p norm, unimodular polynomial, Littlewood polynomial

Mathematical Subject Classification 2010

Primary: 30C10, 11R06

Secondary: 11C08, 60G99

Milestones

Received: 14 June 2010

Accepted: 30 September 2010

Published: 8 October 2011

Proposed: Jonathan D. Rogawski