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=p 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 jn 􀀀 xj x for a xed and x ! 1. This work resolves a function eld analogue of this problem, in the limit of a large niteeld. More precisely, consider monic f0 2 Fq[T] of degree n and take with 1 > 2n. Then the asymptotic density of polynomials f in the `interval' deg(f 􀀀 f0) n that are of the form f = A2+TB2, A;B 2 Fq[T] is 1 4n 􀀀2n n as q ! 1. This density agrees with the asymptotic density of such monic f's of degree n as q ! 1, 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 coecients: The Galois group is the hyperoctahedral group of order 2nn!.