Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Brian Kintu

TL;DR

This work investigates counting questions for $n$-periodic points of polynomial dynamics modulo primes, focusing on $\varphi_{p,c}(z)=z^{p}+c$ (odd prime degree) and $\varphi_{p-1,c}(z)$ (even degree) and reveals exact and conditional counts: $N_{c}^{(n)}(p)$ is $p$ or $0$ for $\varphi_{p,c}$, while the $n$-periodic point counts for $\varphi_{p-1,c}$ modulo $p$ are in $\{1,2,0\}$ with explicit congruence rules. The paper also analyzes dynamical complexity (showing zero exponential growth rate) and averages and densities of these counts, yielding zero or finite limits and demonstrating independence from $n$ in many cases. Extending beyond dynamics, it connects to arithmetic statistics by bounding numbers of polynomials with primitive Galois groups, counting number fields with bounded discriminants and prescribed Galois groups, and examining class numbers; it culminates in equidistribution results for associated Artin $L$-functions in line with Sato–Tate phenomena. Altogether, the results bridge discrete dynamical counting with number-field statistics, providing asymptotic bounds, density results, and L-function equidistribution within these dynamical families.

Abstract

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.

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

TL;DR

This work investigates counting questions for -periodic points of polynomial dynamics modulo primes, focusing on (odd prime degree) and (even degree) and reveals exact and conditional counts: is or for , while the -periodic point counts for modulo are in with explicit congruence rules. The paper also analyzes dynamical complexity (showing zero exponential growth rate) and averages and densities of these counts, yielding zero or finite limits and demonstrating independence from in many cases. Extending beyond dynamics, it connects to arithmetic statistics by bounding numbers of polynomials with primitive Galois groups, counting number fields with bounded discriminants and prescribed Galois groups, and examining class numbers; it culminates in equidistribution results for associated Artin -functions in line with Sato–Tate phenomena. Altogether, the results bridge discrete dynamical counting with number-field statistics, providing asymptotic bounds, density results, and L-function equidistribution within these dynamical families.

Abstract

In this follow-up paper, we inspect a surprising relationship between the set of -periodic points of a polynomial map defined by for all and the coefficient , where is an integer and 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 -torsion point-counting in arithmetic statistics, and also by Hutz's conjecture along with Panraksa's work on -periodic rational point-counting in arithmetic dynamics. In doing so, we then first prove that for any prime and for any fixed (period) , the average number of distinct -periodic integral points of any modulo is unbounded or zero as tends to infinity. Inspired further by a conjecture of Hutz on any for any prime in arithmetic dynamics, we then also prove that for any fixed (period) , the average number of distinct -periodic integral points of any modulo is or or as . 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 -functions that arise naturally in our polynomial discrete dynamical settings.

Paper Structure

This paper contains 12 sections, 32 theorems, 7 equations.

Table of Contents

  1. Introduction
  2. The Number of $n$-Periodic Integral Points of any Family of Polynomial Maps $\varphi_{p,c}$
  3. On Number of $n$-Periodic Integral Points of any Family of Polynomial Maps $\varphi_{p-1,c}$
  4. Dynamical Complexity of Forward Periodic Orbit Structure of any $\varphi_{p,c}$ and $\varphi_{p-1,c}$
  5. On the Average Number of $n$-Periodic Points of any Polynomial Map $\varphi_{p,c}$ & $\varphi_{p-1,c}$
  6. On the Density of Monic Integer Polynomials $\varphi_{p,c}(x)$ with the Number $N^{(n)}_{c}(p) = p$
  7. On Densities of Monic Integer Polynomials $\varphi_{p-1,c}(x)$ with $M_{c}^{(n)}(p) = 1$ & $M_{c}^{(n)}(p) = 2$
  8. On Density of Integer Monics $\varphi_{p,c}(x)$ with $N_{c}^{(n)}(p) = 0$ and $\varphi_{p-1,c}(x)$ with $M_{c}^{(n)}(p) = 0$
  9. On Number of Monic Polynomials $f\in \mathbb{Z}[x]$ & $g\in \mathbb{Z}[x]$ with Primitive Galois groups
  10. Fields $K_{f}$ and $L_{g}$ with Bounded Absolute Discriminant & Prescribed Galois group
  11. On Number of Algebraic Number fields $K_{f}$ and $L_{g}$ with Prescribed Class Number
  12. On Equidistribution of Families of Artin $L$-Functions induced by Fields $K_{f}$ and $L_{g}$

Key Result

Theorem 1.1

Let $p\geq 3$ be any fixed prime, and $n\geq 2$ be any fixed integer. Let $\varphi_{p, c}$ be defined by $\varphi_{p, c}(z) = z^p+c$ for all $c, z\in\mathbb{Z}$. Then the number of distinct $n$-periodic integral points of any $\varphi_{p,c}$ modulo $p$ is $p$ or zero.

Theorems & Definitions (68)

  • Theorem 1.1
  • Theorem 1.2
  • Conjecture 1.3
  • Conjecture 1.4
  • Corollary 1.5
  • Conjecture 1.6
  • Theorem 1.7
  • Theorem 1.8
  • Theorem 1.9
  • Conjecture 1.10
  • ...and 58 more