A polynomial analogue of Landau's theorem and related problems

TL;DR

This paper extends the understanding of polynomial analogues of Landau's theorem over finite fields, providing asymptotic formulas for counting specific polynomial forms in the limit of large degree and field size.

Contribution

It generalizes previous results to the case where both degree and field size grow, using explicit bounds on generating function coefficients, and applies to related problems involving prime factors of even degree.

Findings

Asymptotic formula for B(n,q) as q^n approaches infinity

Explicit constant K_q with error term O(1/q)

Methods based on bounds of generating function coefficients

Abstract

Recently, an analogue over $\mathbb{F}_q[T]$ 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 $\mathbb{F}_q[T]$ of degree $n$ of the form $A^2+TB^2$, 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 $q^n \to \infty$. More precisely, we prove \begin{equation*} B(n,q) \sim K_q \cdot \binom{n-\frac{1}{2}}{n} \cdot q^n , \qquad q^n \to \infty, \end{equation*} for an explicit constant $K_q = 1+O\left(1/q\right)$. 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.