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For fixed coprime polynomials $U,V \in \mathbb{F}_q [T]$ with degrees ofdifferent parities, and a general polynomial $A \in \mathbb{F}_q [T]$, definethe arithmetic function $S_{U,V} (A)$ to be the number of representations of$A$ of the form $UE^2 + VF^2$ with $E,F \in \mathbb{F}_q [T]$. We study themean and variance of $S_{U,V}$ over short intervals in $\mathbb{F}_q [T]$, andthis can be interpreted as the function field analogue of the mean and varianceof lattice points in thin elliptic annuli, where the scaling factor of theellipses is rational. Our main result is an asymptotic formula for the varianceeven when the length of the interval remains constant relative to the absolutevalue of the centre of the interval. In terms of lattice points, this means weobtain the variance in the so-called ``local'' or ``microscopic'' regime, wherethe area of the annulus remains constant relative to the inner radius. We alsoobtain asymptotic or exact formulas for almost all other lengths of theinterval, and we see some interesting behaviour at the boundary between shortand long intervals. Our approach is that of additive characters and Hankelmatrices that we employed for the divisor function and a restrictedsum-of-squares function in previous work, and we develop further results onHankel matrices in this paper.

Lattice Point Variance in Thin Elliptic Annuli over $\mathbb{F}_q [T]$