Taiwanese Journal of Mathematics

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

Chih-Yun Chuang, Yen-Liang Kuan, and Wei-Chen Yao

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

Article information

Source
Taiwanese J. Math., Advance publication (2020), 21 pages.

Dates
First available in Project Euclid: 24 June 2020

Permanent link to this document
https://projecteuclid.org/euclid.twjm/1592985618

Digital Object Identifier
doi:10.11650/tjm/200602

Subjects
Primary: 11N37: Asymptotic results on arithmetic functions 11E12: Quadratic forms over global rings and fields 11T55: Arithmetic theory of polynomial rings over finite fields

Keywords
binary quadratic forms polynomials over finite fields

Citation

Chuang, Chih-Yun; Kuan, Yen-Liang; Yao, Wei-Chen. Variation of a Theme of Landau--Shanks in Positive Characteristic. Taiwanese J. Math., advance publication, 24 June 2020. doi:10.11650/tjm/200602. https://projecteuclid.org/euclid.twjm/1592985618


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