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

For $B\in {\mathbb{F}}_q[T]$ of degree $2n\ge 2$, consider the number of ways of writing $B=E^2+\gamma F^2$, where $\gamma \in {\mathbb{F}}_q^{\ast }$ is fixed, and $E,F\in {\mathbb{F}}_q[T]$ with $\mathrm{d}\mathrm{e}\mathrm{g}E=n$ and $\mathrm{d}\mathrm{e}\mathrm{g}F=m<n$. We denote this by $S_{\gamma ; m} (B)$\mathrm{.}Weobtainanexactformulaforthevarianceof$S_{\gamma ; m} (B)$ 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 $B=E^2+T{F}^2$.