单临界多项式的Galois–动力学对应

Q3 Mathematics Arnold Mathematical Journal Pub Date : 2021-06-09 DOI:10.1007/s40598-021-00179-7
Robin Zhang
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引用次数: 1

摘要

在与Mordell–Lang猜想证明中使用的阿贝尔变体的扭点的Galois同态性质的类比中,我们描述了Galois群的作用和有理映射的动力学作用之间的对应关系。对于具有有理系数的非线性多项式,相关的动态原子多项式的不可约性是一个方便的标准,尽管我们也验证了当动态原子多项式可约时,在几种情况下会出现对应关系。Morton、Morton–Patel和Vivaldi–Hatjispyros在20世纪90年代初的工作将动态原子多项式的不可约性和伽罗瓦理论性质与有理周期点联系起来;从伽罗瓦-动力学对应关系中,我们导出了单临界多项式的二次周期点的类似结果。这足以从Morton和Krumm在显式Hilbert不可约性中的工作中推导出在指定的有限集之外,具有精确周期5和6的二次多项式的二次周期点的不存在性。
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A Galois–Dynamics Correspondence for Unicritical Polynomials

In an analogy with the Galois homothety property for torsion points of abelian varieties that was used in the proof of the Mordell–Lang conjecture, we describe a correspondence between the action of a Galois group and the dynamical action of a rational map. For nonlinear polynomials with rational coefficients, the irreducibility of the associated dynatomic polynomial serves as a convenient criterion, although we also verify that the correspondence occurs in several cases when the dynatomic polynomial is reducible. The work of Morton, Morton–Patel, and Vivaldi–Hatjispyros in the early 1990s connected the irreducibility and Galois-theoretic properties of dynatomic polynomials to rational periodic points; from the Galois–dynamics correspondence, we derive similar consequences for quadratic periodic points of unicritical polynomials. This is sufficient to deduce the non-existence of quadratic periodic points of quadratic polynomials with exact period 5 and 6, outside of a specified finite set from Morton and Krumm’s work in explicit Hilbert irreducibility.

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来源期刊
Arnold Mathematical Journal
Arnold Mathematical Journal Mathematics-Mathematics (all)
CiteScore
1.50
自引率
0.00%
发文量
28
期刊介绍: The Arnold Mathematical Journal publishes interesting and understandable results in all areas of mathematics. The name of the journal is not only a dedication to the memory of Vladimir Arnold (1937 – 2010), one of the most influential mathematicians of the 20th century, but also a declaration that the journal should serve to maintain and promote the scientific style characteristic for Arnold''s best mathematical works. Features of AMJ publications include: Popularity. The journal articles should be accessible to a very wide community of mathematicians. Not only formal definitions necessary for the understanding must be provided but also informal motivations even if the latter are well-known to the experts in the field. Interdisciplinary and multidisciplinary mathematics. AMJ publishes research expositions that connect different mathematical subjects. Connections that are useful in both ways are of particular importance. Multidisciplinary research (even if the disciplines all belong to pure mathematics) is generally hard to evaluate, for this reason, this kind of research is often under-represented in specialized mathematical journals. AMJ will try to compensate for this.Problems, objectives, work in progress. Most scholarly publications present results of a research project in their “final'' form, in which all posed questions are answered. Some open questions and conjectures may be even mentioned, but the very process of mathematical discovery remains hidden. Following Arnold, publications in AMJ will try to unhide this process and made it public by encouraging the authors to include informal discussion of their motivation, possibly unsuccessful lines of attack, experimental data and close by research directions. AMJ publishes well-motivated research problems on a regular basis.  Problems do not need to be original; an old problem with a new and exciting motivation is worth re-stating. Following Arnold''s principle, a general formulation is less desirable than the simplest partial case that is still unknown.Being interesting. The most important requirement is that the article be interesting. It does not have to be limited by original research contributions of the author; however, the author''s responsibility is to carefully acknowledge the authorship of all results. Neither does the article need to consist entirely of formal and rigorous arguments. It can contain parts, in which an informal author''s understanding of the overall picture is presented; however, these parts must be clearly indicated.
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