Counting the number of $n$-periodic integral points of a discrete dynamical system with applications from arithmetic statistics, IV
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| Format: | Preprint |
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2025
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| _version_ | 1866918270345412608 |
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| author | Kintu, Brian |
| author_facet | Kintu, Brian |
| contents | In this follow-up paper, we inspect a surprising relationship between the set of $n$-periodic 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 and $n\geq 2$ is any fixed integer. As before, we again wish to study counting problems which are inspired by the exciting advances of Bhargava-Shankar-Tsimerman and their collaborators on $n$-torsion point-counting in arithmetic statistics, and also by Hutz's conjecture along with Panraksa's work on $n$-periodic rational point-counting in arithmetic dynamics. In doing so, we then first prove that for any prime $p\geq 3$ and for any fixed (period) $n\in \mathbb{Z}_{\geq 2}$, the average number of distinct $n$-periodic integral points of any $φ_{p, c}$ modulo $p$ is unbounded or zero as $c$ tends to infinity. Inspired further by a conjecture of Hutz on any $φ_{p-1, c}$ for any prime $p\geq 5$ in arithmetic dynamics, we then also prove that for any fixed (period) $n\in \mathbb{Z}_{\geq 2}$, the average number of distinct $n$-periodic integral points of any $φ_{p-1, c}$ modulo $p$ is $1$ or $2$ or $0$ as $c\to \infty$. Finally, we then apply density, polynomial-counting, number field-counting, and Sato-Tate equidistribution results from arithmetic statistics, and thereby obtaining a stream of counting and statistical results on irreducible polynomials, number fields, and Artin $L$-functions that arise naturally in our polynomial discrete dynamical settings. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_08601 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Counting the number of $n$-periodic integral points of a discrete dynamical system with applications from arithmetic statistics, IV Kintu, Brian Number Theory Dynamical Systems In this follow-up paper, we inspect a surprising relationship between the set of $n$-periodic 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 and $n\geq 2$ is any fixed integer. As before, we again wish to study counting problems which are inspired by the exciting advances of Bhargava-Shankar-Tsimerman and their collaborators on $n$-torsion point-counting in arithmetic statistics, and also by Hutz's conjecture along with Panraksa's work on $n$-periodic rational point-counting in arithmetic dynamics. In doing so, we then first prove that for any prime $p\geq 3$ and for any fixed (period) $n\in \mathbb{Z}_{\geq 2}$, the average number of distinct $n$-periodic integral points of any $φ_{p, c}$ modulo $p$ is unbounded or zero as $c$ tends to infinity. Inspired further by a conjecture of Hutz on any $φ_{p-1, c}$ for any prime $p\geq 5$ in arithmetic dynamics, we then also prove that for any fixed (period) $n\in \mathbb{Z}_{\geq 2}$, the average number of distinct $n$-periodic integral points of any $φ_{p-1, c}$ modulo $p$ is $1$ or $2$ or $0$ as $c\to \infty$. Finally, we then apply density, polynomial-counting, number field-counting, and Sato-Tate equidistribution results from arithmetic statistics, and thereby obtaining a stream of counting and statistical results on irreducible polynomials, number fields, and Artin $L$-functions that arise naturally in our polynomial discrete dynamical settings. |
| title | Counting the number of $n$-periodic integral points of a discrete dynamical system with applications from arithmetic statistics, IV |
| topic | Number Theory Dynamical Systems |
| url | https://arxiv.org/abs/2507.08601 |