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February, 2021 Variation of a Theme of Landau--Shanks in Positive Characteristic
Chih-Yun Chuang, Yen-Liang Kuan, Wei-Chen Yao
Taiwanese J. Math. 25(1): 23-43 (February, 2021). DOI: 10.11650/tjm/200602

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 $\mathbf{B}_{\mathcal{D}}(n,q)$.

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Chih-Yun Chuang. Yen-Liang Kuan. Wei-Chen Yao. "Variation of a Theme of Landau--Shanks in Positive Characteristic." Taiwanese J. Math. 25 (1) 23 - 43, February, 2021. https://doi.org/10.11650/tjm/200602

Information

Received: 5 March 2020; Revised: 31 May 2020; Accepted: 16 June 2020; Published: February, 2021
First available in Project Euclid: 24 June 2020

Digital Object Identifier: 10.11650/tjm/200602

Subjects:
Primary: 11E12, 11N37, 11T55

Rights: Copyright © 2021 The Mathematical Society of the Republic of China

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Vol.25 • No. 1 • February, 2021
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