Stable binomials over finite fields

IF 1.2 3区数学 Q1 MATHEMATICS

Finite Fields and Their Applications Pub Date : 2024-10-18 DOI:10.1016/j.ffa.2024.102520

Arthur Fernandes , Daniel Panario , Lucas Reis

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引用次数: 0

Abstract

In this paper, we study stable binomials over finite fields, i.e., irreducible binomials

x^{t} - b \in F_{q} [x]

such that all their iterates are also irreducible over

F_{q}

. We obtain a simple criterion on the stability of binomials based on the forward orbit of 0 under the map

z \mapsto z^{t} - b

. In particular, our criterion extends the one obtained by Jones and Boston (2011) for the quadratic case. As applications of our main result, we obtain an explicit 1-parameter family of stable quartics over prime fields

F_{p}

with

p \equiv 5 (\mod 24)

and also develop an algorithm to test the stability of binomials over finite fields. Finally, building upon a work of Ostafe and Shparlinski (2010), we employ character sums to bound the complexity of such algorithm.

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有限域上的稳定二项式

本文研究有限域上的稳定二项式，即不可约二项式 xt-b∈Fq[x]，使得它们的所有迭代也都是 Fq 上的不可约二项式。我们根据 0 在 z↦zt-b 映射下的前向轨道，得到了一个关于二项式稳定性的简单判据。特别是，我们的判据扩展了 Jones 和 Boston（2011）在二次情况下得到的判据。作为我们主要结果的应用，我们得到了质域 Fp 上 p≡5(mod24)的稳定四元数的一个明确的 1 参数族，还开发了一种算法来检验有限域上二项式的稳定性。最后，在 Ostafe 和 Shparlinski（2010 年）工作的基础上，我们利用特征和来约束这种算法的复杂性。

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.