1. Variation of a Theme of Landau--Shanks in Positive Characteristic
- Author
-
Chih-Yun Chuang, Yen-Liang Kuan, and Wei-Chen Yao
- Subjects
Polynomial ,Degree (graph theory) ,General Mathematics ,Polynomial ring ,binary quadratic forms ,Order (ring theory) ,11E12 ,Quadratic form (statistics) ,Combinatorics ,11N37 ,Finite field ,11T55 ,Discriminant ,polynomials over finite fields ,Binary quadratic form ,Mathematics - 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 $\mathbf{B}_{\mathcal{D}}(n,q)$.
- Published
- 2021