arXiv subjects

Brian Kintu

Publications and source records attributed to Brian Kintu.

7 recordsLinked to original sources

arXiv2026

Counting the number of $1_{m}$-preperiodic $\mathcal{O}_{K}$-points of a discrete dynamical system with applications from arithmetic statistics, VII

In this follow-up article of a multi-part series on (strictly) preperiodic point-counting, we inspect an astonishing relationship between the set of (strictly) $1_{m}$-preperiodic points of a polynomial map $\varphi_{d, c}$ defined by $\varphi_{d, c}(z) = z^d + c$ for all $c, z \in \mathcal{O}_{K}$ and the coefficient $c$, where $K$ is any number field of degree $n\geq 1$, $d>2$ is an integer and $m\in \mathbb{Z}_{\geq 1}$ is any fixed (eventual period). As before, we wish to study counting problems that are inspired by torsion point-counting in arithmetic statistics and (strictly) preperiodic point-counting in arithmetic dynamics. In doing so, we then first prove that for any prime $p\geq 3$ and for any fixed $\ell \in \mathbb{Z}_{\geq 1}$ and fixed (eventual period) $m\in \mathbb{Z}_{\geq 1}$, the average number of distinct $1_{m}$-preperiodic integral points of any odd degree map $\varphi_{p^{\ell}, c}$ modulo prime ideal $p\mathcal{O}_{K}$ is unbounded or zero as $c$ tends to infinity. Inspired further by work of Doyle-Poonen, along with conjectural work of Hutz and $\textit{abc}(\textit{d})$-conditional work of Panraksa on $K$-rational preperiodic points of any even degree map $\varphi_{(p-1)^{\ell}, c}$ for any prime $p\geq 5$ in arithmetic dynamics, we then also prove that for any fixed (eventual period) $m \in \mathbb{Z}_{ \geq 1}$, the average number of distinct $1_{m}$-preperiodic integral points of any $\varphi_{(p-1)^{\ell}, c}$ modulo prime ideal $p\mathcal{O}_{K}$ is unbounded or zero as $c\to \infty$. Finally, we then apply density, polynomial- and number field-counting, and Sato-Tate equidistribution results from arithmetic statistics, and thereby obtaining further a stream of counting and statistical results on arithmetic objects that arise naturally in our polynomial discrete dynamical settings.

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arXiv2025

Counting the number of $n$-periodic $\mathbb{Z}_{p}$-and $\mathbb{F}_{p}[t]$-points of a discrete dynamical system with applications from arithmetic statistics, VI

In this follow-up paper, we again inspect a surprising relationship between the set of $n$-periodic points of a polynomial map $\varphi_{d, c}$ defined by $\varphi_{d, c}(z) = z^d + c$ for all $c, z \in \mathbb{Z}_{p}$ or $\in \mathbb{F}_{p}[t]$ and the coefficient $c$, where $d>2$ is an integer and $n\in \mathbb{Z}_{\geq 2}$ is any fixed (period). As before, we study counting problems that are inspired by $n$-torsion point-counting in arithmetic statistics and $n$-periodic point-counting in arithmetic dynamics. In doing so, we then first prove that for any prime $p\geq 3$ and any fixed $\ell \in \mathbb{Z}_{\geq 1}$, the average number of distinct $n$-periodic $p$-adic integral points of any $\varphi_{p^{\ell}, c}$ modulo $p\mathbb{Z}_{p}$ is unbounded or zero as $c\to \infty$; and also prove that for any prime $p\geq 5$, the average number of distinct $n$-periodic $p$-adic integral points of any $\varphi_{(p-1)^{\ell}, c}$ modulo $p\mathbb{Z}_{p}$ is $1$ or $2$ or $0$ as $c\to \infty$. Inspired further by periodic $\mathbb{F}_{p}(t)$-point-counting in arithmetic dynamics, we then also prove that for any prime $p\geq 3$ and any fixed $\ell \in \mathbb{Z}_{\geq 1}$, the average number of distinct $n$-periodic points of any $\varphi_{p^{\ell}, c}$ modulo prime $\pi$ is unbounded or zero as $c$ varies; and also prove that for any prime $p\geq 5$, the average number of distinct $n$-periodic points of any $\varphi_{(p-1)^{\ell}, c}$ modulo $\pi$ is $1$ or $2$ or $0$ as $c$ varies. Finally, we apply density, polynomial-and field-counting, equidistribution results from arithmetic statistics, and then obtain counting and statistical results on irreducible polynomials, (Artin-Mazur) zeta functions, global fields, and on (Artin) $L$-functions arising naturally in our polynomial discrete dynamical settings.

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arXiv2025

Counting the number of $m$-periodic $\mathcal{O}_{K}$-points of a discrete dynamical system with applications from arithmetic statistics, V

In this follow-up paper, we again inspect a surprising relationship between the set of $m$-periodic points of a polynomial map $\varphi_{d, c}$ defined by $\varphi_{d, c}(z) = z^d + c$ for all $c, z \in \mathcal{O}_{K}$ and the coefficient $c$, where $K$ is any number field of degree $n\geq 2$, $d>2$ is an integer and $m\in \mathbb{Z}_{\geq 2}$ is any fixed (period). As before, we again study counting problems which are inspired by advances on $m$-torsion point-counting in arithmetic statistics and $m$-periodic point-counting in arithmetic dynamics. In doing so, we then first prove that for any prime $p\geq 3$ and for any fixed $\ell\in \mathbb{Z}_{ \geq 1}$ and (period) $m\in \mathbb{Z}_{\geq 2}$, the average number of distinct $m$-periodic integral points of any $\varphi_{p^{\ell}, c}$ modulo prime ideal $p\mathcal{O}_{K}$ is unbounded or zero as $c$ tends to infinity. Motivated further by $K$-rational periodic point-counting work of Benedetto along with conjectural work of Hutz on $m$-periodic points of any $\varphi_{(p-1)^{\ell}, c}$ for any prime $p\geq 5$ and any fixed $\ell \in \mathbb{Z}_{\geq 1}$ in arithmetic dynamics, we then also prove that for any fixed (period) $m\in \mathbb{Z}_{\geq 2}$, the average number of distinct $m$-periodic integral points of any $\varphi_{(p-1)^{\ell}, c}$ modulo prime $p\mathcal{O}_{K}$ is $1$ or $2$ or $0$ as $c\to \infty$. Finally, we then apply here density, polynomial-counting, field-counting, and Sato-Tate equidistribution results from arithmetic statistics, and thereby obtaining further counting and statistical results on the irreducible monic polynomials, Artin-Mazur zeta functions, algebraic number fields, and lastly on Artin $L$-functions arising naturally in our polynomial discrete dynamical settings.

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arXiv2025

Counting the number of $n$-periodic integral points of a discrete dynamical system with applications from arithmetic statistics, IV

In this follow-up paper, we inspect a surprising relationship between the set of $n$-periodic points of a polynomial map $\varphi_{d, c}$ defined by $\varphi_{d, c}(z) = z^d + c$ 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 $\varphi_{p, c}$ modulo $p$ is unbounded or zero as $c$ tends to infinity. Inspired further by a conjecture of Hutz on any $\varphi_{p-1, c}$ 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 $\varphi_{p-1, c}$ 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.

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arXiv2025

Counting the number of $\mathbb{Z}_{p}$-and $\mathbb{F}_{p}[t]$-fixed points of a discrete dynamical system with applications from arithmetic statistics, III

In this follow-up paper, we again inspect a surprising relationship between the set of fixed points of a polynomial map $\varphi_{d, c}$ defined by $\varphi_{d, c}(z) = z^d + c$ for all $c, z \in \mathcal{O}_{K}$ or $\in \mathbb{Z}_{p}$ or $\in \mathbb{F}_{p}[t]$ and the coefficient $c$, where $K$ is any number field of degree $n > 1$, $p>2$ is any prime, $\mathbb{Z}_{p}$ (resp., $\mathbb{F}_{p}[t]$) is the ring of all $p$-adic integers (resp., the ring of all polynomials over a finite field $\mathbb{F}_{p}$) and $d>2$ is an integer. As before, we again wish to study counting problems which are inspired by advances in arithmetic statistics, and also by Narkiewicz on totally complex $K$-periodic points along with Adam-Fares on $\mathbb{Q}_{p}$-periodic points in arithmetic dynamics. In doing so, we then first prove that for any prime $p\geq 3$ and for any $\ell \in \mathbb{Z}_{\geq 1}$, the average number of distinct fixed points of any $\varphi_{p^{\ell}, c}$ modulo prime $p\mathcal{O}_{K}$ (modulo $p\mathbb{Z}_{p}$) is bounded or zero or unbounded as $c\to \infty$ . Motivated further by $\mathbb{F}_{p}(t)$-periodic point-counting result of Benedetto in arithmetic dynamics, we then also find that the average number of fixed points in $\mathbb{F}_{p}[t]$-setting behaves in the same way as in $\mathcal{O}_{K}$-setting. Finally, we then apply here counting and statistical results from arithmetic statistics, and as a result obtain counting and statistical results on irreducible monic ($p$-adic) integer polynomials, number fields and subfields of global function fields arising naturally in our polynomial discrete dynamical settings.

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arXiv2025

Counting the number of $\mathcal{O}_{K}$-fixed points of a discrete dynamical system with applications from arithmetic statistics, II

In this follow-up paper, we again inspect a surprising connection between the set of fixed points of a polynomial map $\varphi_{d,c}$ defined by $\varphi_{d,c}(z) = z^d + c$ for all $c, z \in \mathcal{O}_{K}$ and the coefficient $c$, where $K$ is any number field of degree $n > 1$ and $d > 2$ is an integer. As before, we wish to study counting problems which are inspired by exciting advances in arithmetic statistics, and again partly by point-counting result of Narkiewicz on real $K$-rational periodic points of any odd degree map $\varphi_{d,c}$ in arithmetic dynamics. In doing so, we then first prove that for any real algebraic number field $K$ of degree $n \geq 2$, and for any prime $p \geq 3$ and integer $\ell \geq 1$, the average number of distinct integral fixed points of any $\varphi_{p^{\ell},c}$ modulo prime ideal $p\mathcal{O}_{K}$ is $3$ or $0$ as $c\to \infty$. Motivated further by $K$-rational periodic point-counting result of Benedetto on any $\varphi_{(p-1)^{\ell},c}$ for any prime $p \geq 5$ and integer $\ell \in \mathbb{Z}_{\geq 1}$ in arithmetic dynamics, we then also prove unconditionally that for any number field (not necessarily real) $K$ of degree $n \geq 2$, the average number of distinct integral fixed points of any $\varphi_{(p-1)^{\ell},c}$ modulo prime $p\mathcal{O}_{K}$ 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 irreducible polynomials and number fields arising naturally in our polynomial discrete dynamical settings.

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arXiv2024

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

In this first article of a multi-part series, we inspect a surprising relationship between the set of fixed points of a polynomial map $\varphi_{d, c}$ defined by $\varphi_{d, c}(z) = z^d + c$ 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 $\varphi_{d, c}$ in arithmetic dynamics, we then first prove that for any prime $p\geq 3$, the average number of distinct integral fixed points of any $\varphi_{p, c}$ modulo $p$ is $3$ or $0$ as $c$ tends to infinity. Inspired further by a conjecture of Hutz on rational periodic points of $\varphi_{p-1, c}$ for any prime $p\geq 5$ in arithmetic dynamics, we then also prove that the average number of distinct integral fixed points of any $\varphi_{p-1, c}$ 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.

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