Vol. 263, No. 2, 2013

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ISSN: 0030-8730
Asymptotic L4 norm of polynomials derived from characters

Daniel J. Katz

Vol. 263 (2013), No. 2, 373–398
Abstract

Littlewood investigated polynomials with coefficients in {−1,1} (Littlewood polynomials), to see how small their ratio of norms f4f2 on the unit circle can become as deg(f) →∞. A small limit is equivalent to slow growth in the mean square autocorrelation of the associated binary sequences of coefficients of the polynomials. The autocorrelation problem for arrays and higher dimensional objects has also been studied; it is the natural generalization to multivariable polynomials. Here we find, for each n > 1, a family of n-variable Littlewood polynomials with lower asymptotic f4f2 than any known hitherto. We discover these through a wide survey, infeasible with previous methods, of polynomials whose coefficients come from finite field characters. This is the first time that the lowest known asymptotic ratio of norms f4f2 for multivariable polynomials f(z1,,zn) is strictly less than what could be obtained by using products f1(z1)fn(zn) of the best known univariate polynomials.

Keywords
L4 norm, Littlewood polynomial, character polynomial, Fekete polynomial, character sum
Mathematical Subject Classification 2010
Primary: 11C08
Secondary: 11T24, 42A05, 11B83
Milestones
Received: 13 June 2012
Accepted: 16 October 2012
Published: 31 May 2013
Authors
Daniel J. Katz
Department of Mathematics
Simon Fraser University
8888 University Drive
Burnaby BC V5A 1S6
Canada