Nilpotent linearized polynomials over finite fields, revisited

IF 1.2 3区数学 Q1 MATHEMATICS Finite Fields and Their Applications Pub Date : 2024-05-08 DOI:10.1016/j.ffa.2024.102442

Daniel Panario , Lucas Reis

引用次数: 0

Abstract

In this paper we develop further studies on nilpotent linearized polynomials (NLP's) over finite fields, a class of polynomials introduced by the second author. We characterize certain NLP's that are binomials and show that, in general, NLP's are also nilpotent over a particular tower of finite fields. We also develop results on the construction of permutation polynomials from NLP's, extending some past results. In particular, the latter yields polynomials that permutes certain infinite subfields of ${\overline{F}}_{q}$ and have a very particular cycle structure. Finally, we provide a nice correspondence between certain NLP's and involutions in binary fields and, in particular, we discuss a general method to produce affine involutions over binary fields without fixed points.

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有限域上的无势线性化多项式，再论

在本文中，我们进一步研究了有限域上的零势线性化多项式（NLP），这是第二位作者提出的一类多项式。我们描述了某些二项式 NLP 的特征，并证明一般来说，有限域上的 NLP 也是零势的。我们还发展了从 NLP 构建置换多项式的结果，扩展了过去的一些结果。特别是，后者产生的多项式可以对 F‾q 的某些无限子域进行置换，并具有非常特殊的循环结构。最后，我们提供了二元域中某些 NLP 与渐开线之间的良好对应关系，特别是，我们讨论了在二元域上产生无定点仿射渐开线的一般方法。

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.

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