Intersection of orbits for polynomials in characteristic p

IF 0.6 3区数学 Q3 MATHEMATICS

Journal of Number Theory Pub Date : 2025-06-06 DOI:10.1016/j.jnt.2025.04.019

Simone Coccia , Dragos Ghioca , Jungin Lee , GyeongHyeon Nam

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引用次数: 0

Abstract

In [GTZ08], [GTZ12], the following result was established: given polynomials

f, g \in C [x]

of degrees larger than 1, if there exist

α, β \in C

such that their corresponding orbits

O_{f} (α)

and

O_{g} (β)

(under the action of f, respectively of g) intersect in infinitely many points, then f and g must share a common iterate, i.e.,

f^{m} = g^{n}

for some

m, n \in N

. If one replaces

C

with a field K of characteristic p, then the conclusion fails; we provide numerous examples showing the complexity of the problem over a field of positive characteristic. We advance a modified conjecture regarding polynomials f and g which admit two orbits with infinite intersection over a field of characteristic p. Then we present various partial results, along with connections with another deep conjecture in the area, the dynamical Mordell-Lang conjecture.

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特征p中多项式的轨道相交

在[GTZ08]， [GTZ12]中，建立了如下结果：给定阶数大于1的多项式f，g∈C[x]，如果存在α，β∈C，使得它们对应的轨道of （α）和Og(β)（分别在g的f作用下）相交于无穷多个点，则f和g必须有一个公共迭代，即对于某m，n∈n, fm=gn。如果用特征为p的域K代替C，则结论不成立；我们提供了大量的例子来说明问题的复杂性。我们提出了一个关于多项式f和g的修正猜想，它允许在特征为p的场上有无限相交的两个轨道。然后我们给出了不同的部分结果，以及与该区域内另一个深度猜想——动态莫德尔-朗猜想的联系。

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

Journal of Number Theory 数学-数学

CiteScore

1.30

自引率

14.30%

发文量

122

审稿时长

16 weeks

期刊介绍： The Journal of Number Theory (JNT) features selected research articles that represent the broad spectrum of interest in contemporary number theory and allied areas. A valuable resource for mathematicians, the journal provides an international forum for the publication of original research in this field. The Journal of Number Theory is encouraging submissions of quality, long articles where most or all of the technical details are included. The journal now considers and welcomes also papers in Computational Number Theory. Starting in May 2019, JNT will have a new format with 3 sections: JNT Prime targets (possibly very long with complete proofs) high impact papers. Articles published in this section will be granted 1 year promotional open access. JNT General Section is for shorter papers. We particularly encourage submission from junior researchers. Every attempt will be made to expedite the review process for such submissions. Computational JNT . This section aims to provide a forum to disseminate contributions which make significant use of computer calculations to derive novel number theoretic results. There will be an online repository where supplementary codes and data can be stored.