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arXiv 2501.04026math.NT
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计数离散动力系统整数不动点的数量及其在算术统计中的应用，I

Counting the number of integral fixed points of a discrete dynamical system with applications from arithmetic statistics, I

University of Toronto（多伦多大学）

Brian Kintu

备注：
13 pages, rewritten to make more insightful, and any comments are very welcome!
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AI总结：

本文研究了离散动力系统整数不动点的计数问题，通过算术统计方法得出了关于不同素数下不动点数量的统计结果。

摘要：

在这一多部分系列的第一篇文章中，我们探讨了由φ_{d,c}(z)=z^d +c定义的多项式映射的不动点集与系数c之间的惊人关系，其中d>2为整数。受Bhargava-Shankar-Tsimerman及其同事在算术统计中优雅计数问题和非常引人注目的结果的启发，以及Narkiewicz关于任意奇数次映射φ_{d,c}在算术动力学中有趣点计数结果的启发，我们首先证明对于任何素数p≥3，当c趋于无穷时，任何φ_{p,c}模p的不同的整数不动点的平均数量为3或0。受Hutz关于任意素数p≥5的φ_{p-1,c}有理周期点的猜想的启发，我们还证明了当c趋于无穷时，任何φ_{p-1,c}模p的不同的整数不动点的平均数量为1或2或0。最后，我们应用算术统计中的密度和数域计数结果，并因此获得了在我们多项式离散动力系统中自然出现的不可约整数多项式和数域的计数和统计结果。

英文摘要：

In this first article of a multi-part series, we inspect a surprising relationship between the set of fixed points of a polynomial map $φ_{d, c}$ defined by $φ_{d, c}(z) = z^d + c$ for all $c, z \in \mathbb{Z}$ and the coefficient $c$, where $d > 2$ is an integer. Inspired greatly by the elegant counting problems along with the very striking results of Bhargava-Shankar-Tsimerman and their collaborators in arithmetic statistics, and also by interesting point-counting result of Narkiewicz on rational periodic points of any odd degree map $φ_{d, c}$ in arithmetic dynamics, we then first prove that for any prime $p\geq 3$, the average number of distinct integral fixed points of any $φ_{p, c}$ modulo $p$ is $3$ or $0$ as $c$ tends to infinity. Inspired further by a conjecture of Hutz on rational periodic points of $φ_{p-1, c}$ for any prime $p\geq 5$ in arithmetic dynamics, we then also prove that the average number of distinct integral fixed points of any $φ_{p-1, c}$ modulo $p$ is $1$ or $2$ or $0$ as $c\to \infty$. Finally, we then apply density and number field-counting results from arithmetic statistics, and as a result obtain counting and statistical results on the irreducible integer polynomials and number fields arising naturally in our polynomial discrete dynamical settings.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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