Variation of a Theme of Landau--Shanks in Positive Characteristic

Abstract

Let $\mathbf{A} := \mathbb{F}_q[t]$ be a polynomial ring over a finite field $\mathbb{F}_q$ of odd characteristic and let $D \in \mathbf{A}$ be a square-free polynomial. Denote by $\mathbf{N}_{D}(n,q)$ the number of polynomials $f$ in $\mathbf{A}$ of degree $n$ which may be represented in the form $u \cdot f = A^2-DB^2$ for some $A,B \in \mathbf{A}$ and $u \in \mathbb{F}_q^{\times}$, and by $\mathbf{B}_{\mathcal{D}}(n,q)$ the number of polynomials in $\mathbf{A}$ of degree $n$ which can be represented by a primitive quadratic form of a given discriminant $\mathcal{D} \in \mathbf{A}$, not necessary square-free. If the class number of the maximal order of $\mathbb{F}_q(t,\sqrt{D})$ is one, then we give very precise asymptotic formulas for $\mathbf{N}_{D}(n,q)$. Moreover, we also give very precise asymptotic formulas for $\mathbf{B}_{\mathcal{D}}(n,q)$.