On the stopping time of the Collatz map in F2[x]

IF 1.2 3区数学 Q1 MATHEMATICS

Finite Fields and Their Applications Pub Date : 2024-08-01 DOI:10.1016/j.ffa.2024.102473

Gil Alon , Angelot Behajaina , Elad Paran

引用次数: 0

Abstract

We study the stopping time of the Collatz map for a polynomial $f \in F_{2} [x]$ , and bound it by $O (\deg {(f)}^{1.5})$ , improving upon the quadratic bound proven by Hicks, Mullen, Yucas and Zavislak. We also prove the existence of arithmetic sequences of unbounded length in the stopping times of certain sequences of polynomials, a phenomenon observed in the classical Collatz map.

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关于 F2[x] 中科拉茨图的停止时间

我们研究了多项式 f∈F2[x] 的科拉茨映射停止时间，并将其约束为 O(deg(f)1.5)，改进了希克斯、马伦、尤卡斯和扎维斯拉克证明的二次约束。我们还证明了在某些多项式序列的停止时间中存在长度无界的算术序列，这是在经典科拉茨图中观察到的现象。

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.