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 ≤ n ≤ x is K/ √ 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| ≤ x ǫ for a fixed ǫ and x → ∞.This work resolves a function field analogue of this problem, in the limit of a large finite field. More precisely, consider monic f 0 ∈ F q [T ] of degree n and take ǫ with 1 > ǫ ≥ 2 n . Then the asymptotic density of polynomials f in the 'interval' deg(fn as q → ∞. This density agrees with the asymptotic density of such monic f 's of degree n as q → ∞, 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!.

The variance of the number of sums of two squares in $\mathbb{F}_q[T]$ in short intervals

“…Remark 2. Theorem 1.1 complements work in [BBSF18], which shows that in all short intervals with h ≥ 2, the count ν α (A; h) is asymptotic to the mean as q → ∞.…”

Chebotarev density theorem in short intervals for extensions of 𝔽_{𝕢}(𝕋)

“…Remark 6.3. Proposition 4.6 in [BBSF15] coincides with the special case of Proposition 6.2 where G = Z/2Z, C = A 1 and π(x) = x 2 . Cyclic extensions with genus 0 were partly treated by Cohen [Coh80, Thm.…”

Squarefree polynomials with prescribed coefficients

“…The function field analog of our setup, where the digits are replaced by the coefficients of a polynomial (1.3) m(T ) = a 0 + a 1 T + · · · + a n−1 T n−1 + a n T n over a finite field F q , has also received a lot of attention, as can be seen from [1,3,4,10,11,14,15,21,23,24,25,26,29,30,31,32,33,38,39,41,42,43,44,45,46,47,48,53,54,55,57]. In this work we contribute to the study of the function field analog by giving lower bounds on the number of squarefrees in 'sparse' boxes.…”

“…over a finite field F q , has also received a lot of attention, as can be seen from [1,3,4,10,11,14,15,21,23,24,25,26,29,30,31,32,33,38,39,41,42,43,44,45,46,47,48,53,54,55,57]. In this work we contribute to the study of the function field analog by giving lower bounds on the number of squarefrees in 'sparse' boxes.…”