Fagundes-Mello conjecture over a finite field

IF 1.2 3区数学 Q1 MATHEMATICS

Finite Fields and Their Applications Pub Date : 2025-03-25 DOI:10.1016/j.ffa.2025.102620

Qian Chen , Yingyu Luo , Yu Wang

引用次数: 0

Abstract

The Fagundes-Mello conjecture asserts that the image of every multilinear polynomial on upper triangular matrix algebras over a field is a vector space, which is an important variation of the old and famous Lvov-Kaplansky conjecture. The Fagundes-Mello conjecture has been solved over an infinite field by Gargate and Mello in 2022. In the present paper, we shall give a result on evaluations of homogenous polynomials with commutative variables over a finite field. As an application, we give a positive answer of the Fagundes-Mello conjecture under a mild condition on the ground field, which covers all results on the Fagundes-Mello conjecture.

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有限域上的Fagundes-Mello猜想

Fagundes-Mello猜想认为在一个域上的上三角矩阵代数上的每一个多线性多项式的像都是一个向量空间，这是对著名的Lvov-Kaplansky猜想的一个重要改进。2022年，加盖特和梅洛在无限场上解决了Fagundes-Mello猜想。本文给出了有限域上具有交换变量的齐次多项式的求值的一个结果。作为一个应用，我们给出了Fagundes-Mello猜想在一个温和条件下的一个正解，它涵盖了Fagundes-Mello猜想的所有结果。

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.