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

Abstract

Landau's theorem asserts that the asymptotic density of sums of two squares in the interval $1\leq n\leq x$ is $K/{\sqrt{\log x}}$, where $K$ is the Landau-Ramanujan constant. It is an old problem in number theory whether the asymptotic density remains the same in intervals $|n-x|\leq x^{\epsilon}$ for a fixed $\epsilon$ and $x\to \infty$. 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 $\epsilon$ with $1>\epsilon\geq \frac2n$. Then the asymptotic density of polynomials $f$ in the `interval' ${\rm deg}(f-f_0)\leq \epsilon n$ that are of the form $f=A^2+TB^2$, $A,B\in \mathbb{F}_q[T]$ 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 $q\to \infty$, 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 $2^nn!$.