Littlewood investigated
polynomials with coefficients in {−1,1} (Littlewood polynomials), to see
how small their ratio of norms ∥f∥4∕∥f∥2 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 ∥f∥4∕∥f∥2 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 ∥f∥4∕∥f∥2 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
