The distribution of the L_4 norm of Littlewood polynomials
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Classical conjectures due to Littlewood, Erdős and Golay concern the asymptotic growth of the L_p norm of a Littlewood polynomial (having all coefficients in {1, -1}) as its degree increases, for various values of p. Attempts over more than fifty years to settle these conjectures have identified certain classes of the Littlewood polynomials as particularly important: skew-symmetric polynomials, reciprocal polynomials, and negative reciprocal polynomials. Using only elementary methods, we find an exact formula for the mean and variance of the L_4 norm of polynomials in each of these classes, and in the class of all Littlewood polynomials. A consequence is that, for each of the four classes, the normalized L_4 norm of a polynomial drawn uniformly at random from the class converges in probability to a constant as the degree increases.
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