A polynomial analogue of Landau's theorem and related problems

Recently, an analogue over Fq[T] of Landau’s theorem on sums of two squares was considered by Bary-Soroker, Smilansky and Wolf. They counted the number of monic polynomials in Fq[T] of degree n of the form A2 + T B2, which we denote by B(n, q). They studied B(n, q) in two limits: fixed n and large q; and fixed q and large n. We generalize their result to the most general limit qn → ∞. More precisely, we prove B(n, q) ∼ Kq ·n −12n!· qn, qn → ∞, for an explicit constant Kq = 1 + O (1/q). Our methods are different and are based on giving explicit bounds on the coefficients of generating functions. These methods also apply to other problems, related to polynomials with prime factors of even degree