The Variance of the Sum of Two Squares over Intervals in $\mathbb{F}_q [T]$: I

TL;DR

This paper derives an exact formula for the variance of the number of representations of polynomials as sums of two squares over intervals in finite fields, using additive characters and Hankel matrices.

Contribution

It provides a novel exact variance formula for sums of two squares in polynomial rings over finite fields, extending previous methods.

Findings

Exact variance formula derived for sums of two squares.

Method employs additive characters and Hankel matrices.

Potential extension to other quadratic forms discussed.

Abstract

For $B \in \mathbb{F}_q [T]$ of degree $2n \geq 2$, consider the number of ways of writing $B=E^2 + \gamma F^2$, where $\gamma \in \mathbb{F}_q^*$ is fixed, and $E,F \in \mathbb{F}_q [T]$ with $\mathrm{deg} \hspace{0.25em} E = n$ and $\mathrm{deg} \hspace{0.25em} F = m < n$. We denote this by $S_{\gamma ; m} (B)$. We obtain an exact formula for the variance of $S_{\gamma ; m} (B)$ over intervals in $\mathbb{F}_q [T]$. We use the method of additive characters and Hankel matrices that the author previously used for the variance and correlations of the divisor function. In Section 2, we give a short overview of our approach; and we briefly discuss the possible extension of our result to the number of ways of writing $B=E^2 + T F^2$.