On the number of rational points of Artin–Schreier’s curves and hypersurfaces

IF 1.4 2区数学 Q3 COMPUTER SCIENCE, THEORY & METHODS

Designs, Codes and Cryptography Pub Date : 2024-06-05 DOI:10.1007/s10623-024-01431-9

F. E. Brochero Martínez, Daniela Oliveira

引用次数: 0

Abstract

Let \(\mathbb {F}_{q^n}\) represent the finite field with \(q^n\) elements. In this paper, our focus is on determining the number of \(\mathbb {F}_{q^n}\)-rational points for two specific objects: an affine Artin–Schreier curve given by the equation \(y^q-y = x(x^{q^i}-x)-\lambda \), and an Artin–Schreier hypersurface given by the equation \(y^q-y=\sum _{j=1}^r a_jx_j(x_j^{q^{i_j}}-x_j)-\lambda \). Additionally, we establish that the Weil bound is only achieved in these cases when the trace of the element \(\lambda \in \mathbb {F}_{q^n}\) over the subfield \(\mathbb {F}_q\) is equal to zero.

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论阿尔廷-施莱尔曲线和超曲面的有理点数

让 \(\mathbb {F}_{q^n}\) 表示具有 \(q^n\) 元素的有限域。本文的重点是确定两个特定对象的 \(\mathbb {F}_{q^n}\) 有理点的数目：由方程 \(y^q-y = x(x^{q^i}-x)-\lambda\) 给出的仿射阿尔丁-施莱尔曲线，以及由方程 \(y^q-y=\sum _{j=1}^r a_jx_j(x_j^{q^{i_j}}-x_j)-\lambda\) 给出的阿尔丁-施莱尔超曲面。此外，我们还确定，只有当子域 \(\mathbb {F}_{q^n}\) 上的元素 \(\mathbb {F}_q\) 的迹等于零时，Weil 约束才会在这些情况下实现。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Designs, Codes and Cryptography 工程技术-计算机：理论方法

CiteScore

2.80

自引率

12.50%

发文量

157

审稿时长

16.5 months

期刊介绍： Designs, Codes and Cryptography is an archival peer-reviewed technical journal publishing original research papers in the designated areas. There is a great deal of activity in design theory, coding theory and cryptography, including a substantial amount of research which brings together more than one of the subjects. While many journals exist for each of the individual areas, few encourage the interaction of the disciplines. The journal was founded to meet the needs of mathematicians, engineers and computer scientists working in these areas, whose interests extend beyond the bounds of any one of the individual disciplines. The journal provides a forum for high quality research in its three areas, with papers touching more than one of the areas especially welcome. The journal also considers high quality submissions in the closely related areas of finite fields and finite geometries, which provide important tools for both the construction and the actual application of designs, codes and cryptographic systems. In particular, it includes (mostly theoretical) papers on computational aspects of finite fields. It also considers topics in sequence design, which frequently admit equivalent formulations in the journal’s main areas. Designs, Codes and Cryptography is mathematically oriented, emphasizing the algebraic and geometric aspects of the areas it covers. The journal considers high quality papers of both a theoretical and a practical nature, provided they contain a substantial amount of mathematics.