
Sums of two squares in short intervals in polynomial rings over finite fields
 American Journal of Mathematics
 Johns Hopkins University Press
 Volume 140, Number 4, August 2018
 pp. 11131131
 10.1353/ajm.2018.0025
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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 LandauRamanujan constant. It is an old problem in number theory whether the asymptotic density remains the same in intervals $nx\leq x^{\epsilon}$ 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 $f_0\in\Bbb{F}_q[T]$ of degree $n$ and take $\epsilon$ with $1>\epsilon\geq {2\over n}$. Then the asymptotic density of polynomials $f$ in the "interval" $\deg(ff_0)\leq\epsilon n$ that are of the form $f=A^2+TB^2$, $A,B\in\Bbb{F}_q[T]$ is ${1\over 4^n}\big(\matrix{2n\atop n}\big)$ 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 $2^nn!$.