Sums of two squares in short intervals in polynomial rings over finite fields

Landau’s theorem asserts that the asymptotic density of sums of two squares in the interval $1\le n\le x$ is $K\mathrm{/}\sqrt{\log x}$, where $K$ is the Landau-Ramanujan constant. It is an old problem in number theory whether the asymptotic density remains the same in intervals $\mathrm{\mid }n-x\mathrm{\mid }\le x^{ϵ}$ for a fixed $ϵ$ and $x\to \mathrm{\infty }$. This work resolves a function field analogue of this problem, in the limit of a large finite field. More precisely, consider monic $f_0\in {\mathbb{F}}_q[T]$ of degree $n$ and take $ϵ$ with $1>ϵ\ge \frac{2}{n}$. Then the asymptotic density of polynomials $f$ in the `interval’ ${\rm deg}(f-f_0)\leq \epsilon n$thatareoftheform$f=A^2+TB^2$,$A,B\in $\mathbb{F}$_q[T]$is$\frac{1}{4^n}\binom{2n}{n}$as$q\to \infty$. This density agrees with the asymptotic density of such monic $f$’s of degree $n$ as $q\to \mathrm{\infty }$, as was shown by the second author, Smilanski, and Wolf. A key point in the proof is the calculation of the Galois group of $f(-T^2)$, where $f$ is a polynomial of degree $n$ with a few variable coefficients: The Galois group is the hyperoctahedral group of order $2^{n}n!$.