On the set of stable primes for postcritically infinite maps over number fields

IF 1.2 3区数学 Q1 MATHEMATICS

Finite Fields and Their Applications Pub Date : 2025-07-07 DOI:10.1016/j.ffa.2025.102693

Joachim König

引用次数: 0

Abstract

Many interesting questions in arithmetic dynamics revolve, in one way or another, around the (local and/or global) reducibility behavior of iterates of a polynomial. We show that for very general families of integer polynomials f (and, more generally, rational functions over number fields), the set of stable primes, i.e., primes modulo which all iterates of f are irreducible, is a density zero set. Compared to previous results, our families cover a much wider ground, and in particular apply to 100% of polynomials of any given odd degree, thus adding evidence to the conjecture that polynomials with a “large” set of stable primes are necessarily of a very specific shape, and in particular are necessarily postcritically finite.

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数域上后临界无限映射的稳定素数集

算术动力学中许多有趣的问题以某种方式围绕多项式迭代的（局部和/或全局）可约性行为。我们证明了对于非常一般的整数多项式族f（以及更一般的数域上的有理函数），稳定素数的集合，即所有迭代f都不可约的素数模，是一个密度零集。与以前的结果相比，我们的家族涵盖了更广泛的领域，特别是适用于任何给定奇数次的多项式的100%，从而为猜想提供了证据，即具有“大”稳定素数集的多项式必然具有非常特定的形状，特别是必然是后临界有限的。

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.