The variance of a restricted sum-of-squares function over short intervals in [formula omitted]

Abstract

For B∈Fq[T] of degree 2n≥2, consider the number of ways of writing B=E2+γF2, where γ∈Fq⁎ is fixed, and E,F∈Fq[T] with degE=n and degF=m<n. We denote this restricted sum-of-squares function by Sγ;m(B). We obtain an exact formula for the variance of Sγ;m(B) over short intervals in Fq[T] where q is an odd prime power. Interestingly, when m+1≤n≤2m, the variance vanishes on some (not all) short intervals, which is in contrast to the standard function that counts representations as a sum of two squares. We use the method of additive characters and Hankel matrices that we previously used for the variance and correlations of the divisor function. In Section 2, we give a short overview of our approach. Section 3 gives new results relating to values of quadratic forms, over Fq, that are defined from Hankel matrices.