Vol. 252, No. 1, 2011

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Average Mahler’s measure and Lp norms of unimodular polynomials

Kwok-Kwong Stephen Choi and Michael J. Mossinghoff

Vol. 252 (2011), No. 1, 31–50

A polynomial f [z] is unimodular if all its coefficients have unit modulus. Let Un denote the set of unimodular polynomials of degree n1, and let Un denote the subset of reciprocal unimodular polynomials, which have the property that f(z) = ωzn1f(1z) 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)√n- and L p norm ||f||p√n- over the sets Un and Un, 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 eγ∕2 = 0.749306 as n →∞ for unimodular polynomials, and eγ∕2√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 Lp norm very close to the respective limiting mean value.

mean Mahler’s measure, mean Lp norm, unimodular polynomial, Littlewood polynomial
Mathematical Subject Classification 2010
Primary: 30C10, 11R06
Secondary: 11C08, 60G99
Received: 14 June 2010
Accepted: 30 September 2010
Published: 8 October 2011

Proposed: Jonathan D. Rogawski
Kwok-Kwong Stephen Choi
Department of Mathematics
Simon Fraser University
Burnaby, British Columbia V5A 1S6
Michael J. Mossinghoff
Department of Mathematics
Davidson College
Davidson, North Carolina 28035-6996
United States