A correction and further improvements to the Chevalley-Warning theorems

IF 1.2 3区数学 Q1 MATHEMATICS

Finite Fields and Their Applications Pub Date : 2024-04-08 DOI:10.1016/j.ffa.2024.102427

David B. Leep , Rachel L. Petrik

引用次数: 0

Abstract

This paper corrects an error in the proof of Theorem 1.4 (3) of our earlier paper, Further Improvements to the Chevalley-Warning Theorems. The error originally appeared in Heath-Brown's paper, On Chevalley-Warning Theorems, which invalidates the proof of Theorem 2 (iii) in that paper. In this paper, we use a new method to give a correct proof of Theorem 1.4 (3). The correction in this paper also fixes the proof of Theorem 2 (iii) in Heath-Brown's paper. The proof in this paper provides slightly stronger estimates for some of the inequalities that were used in Further Improvements to the Chevalley-Warning Theorems.

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对切瓦利-沃宁定理的修正和进一步改进

本文纠正了我们早先的论文《对车瓦利警告定理的进一步改进》中定理 1.4 (3) 证明中的一个错误。这个错误最初出现在希斯-布朗（Heath-Brown）的论文《论车瓦利-警告定理》中，导致该论文中定理 2 (iii) 的证明无效。在本文中，我们用一种新方法给出了定理 1.4 (3) 的正确证明。本文的修正也修正了希斯-布朗论文中定理 2 (iii) 的证明。本文的证明为《对切瓦利-警告定理的进一步改进》中使用的一些不等式提供了稍强的估计。

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

Finite Fields and Their Applications 数学-数学

CiteScore

2.00

自引率

20.00%

发文量

133

审稿时长

6-12 weeks

期刊介绍： Finite Fields and Their Applications is a peer-reviewed technical journal publishing papers in finite field theory as well as in applications of finite fields. As a result of applications in a wide variety of areas, finite fields are increasingly important in several areas of mathematics, including linear and abstract algebra, number theory and algebraic geometry, as well as in computer science, statistics, information theory, and engineering. For cohesion, and because so many applications rely on various theoretical properties of finite fields, it is essential that there be a core of high-quality papers on theoretical aspects. In addition, since much of the vitality of the area comes from computational problems, the journal publishes papers on computational aspects of finite fields as well as on algorithms and complexity of finite field-related methods. The journal also publishes papers in various applications including, but not limited to, algebraic coding theory, cryptology, combinatorial design theory, pseudorandom number generation, and linear recurring sequences. There are other areas of application to be included, but the important point is that finite fields play a nontrivial role in the theory, application, or algorithm.