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 ≤ n ≤ x is K/ √logx, 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^ϵ for a fixed ϵ and x → ∞. This work resolves a function field analogue of this problem, in the limit of a large finite field. More precisely, consider monic f₀ ∈ F_q [T] of degree n and take ϵ with 1 > ϵ ≥ 2/n. Then the asymptotic density of polynomials f in the “interval” deg(f − f₀) ≤ ϵn that are of the form f = A² +TB², A,B ∈ F_q [T] is 1/4n (2n/n) 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(−T²), where f is a polynomial of degree n with a few variable coefficients: The Galois group is the hyperoctahedral group of order 2ⁿn!.