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

TL;DR

This paper investigates the distribution of sums of two squares within short polynomial intervals over finite fields, establishing an asymptotic density that aligns with classical number theory results in a function field setting.

Contribution

It provides a function field analogue of Landau's theorem for sums of two squares in short intervals, including the calculation of the Galois group of certain polynomials.

Findings

Asymptotic density of sums of two squares matches classical predictions

Galois group of specific polynomials is the hyperoctahedral group

Density results hold as the size of the finite field tends to infinity

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 $2^nn!$.