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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

Brian Kintu

TL;DR

This work develops a comprehensive fixed-point counting framework for polynomial dynamical systems of the form $\varphi_{d,c}(z)=z^d+c$ across multiple local settings: $\mathcal{O}_{K}$, $\mathbb{Z}_{p}$, and $\mathbb{F}_{p}[t]$, including reductions modulo primes, primes in inert positions in number fields, and moduli defined by irreducible polynomials in $\mathbb{F}_{p}[t]$. The authors establish exact dichotomies for the number of fixed points modulo these bases (e.g., $p$ or $0$ in the $p$-adic and integral settings, and $p$ or $0$ modulo $\pi$ in function fields), and they derive precise average/count-density statements as the coefficients $c$ grow or as degrees vary. The paper then connects these local dynamics to global arithmetic statistics, yielding bounds and asymptotics for counts of irreducible polynomials, number fields, and subfields in corresponding dynamical families, leveraging recent results of Bhargava–Shankar–Wang and Lenstra on densities and maximal orders. Collectively, the results illuminate how fixed-point data in local dynamical systems reflects global field-theoretic structures and yield quantitative insights into field counting and monogenicity in arithmetic statistics contexts.

Abstract

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.

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

TL;DR

This work develops a comprehensive fixed-point counting framework for polynomial dynamical systems of the form across multiple local settings: , , and , including reductions modulo primes, primes in inert positions in number fields, and moduli defined by irreducible polynomials in . The authors establish exact dichotomies for the number of fixed points modulo these bases (e.g., or in the -adic and integral settings, and or modulo in function fields), and they derive precise average/count-density statements as the coefficients grow or as degrees vary. The paper then connects these local dynamics to global arithmetic statistics, yielding bounds and asymptotics for counts of irreducible polynomials, number fields, and subfields in corresponding dynamical families, leveraging recent results of Bhargava–Shankar–Wang and Lenstra on densities and maximal orders. Collectively, the results illuminate how fixed-point data in local dynamical systems reflects global field-theoretic structures and yield quantitative insights into field counting and monogenicity in arithmetic statistics contexts.

Abstract

In this follow-up paper, we again inspect a surprising relationship between the set of fixed points of a polynomial map defined by for all or or and the coefficient , where is any number field of degree , is any prime, (resp., ) is the ring of all -adic integers (resp., the ring of all polynomials over a finite field ) and 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 -periodic points along with Adam-Fares on -periodic points in arithmetic dynamics. In doing so, we then first prove that for any prime and for any , the average number of distinct fixed points of any modulo prime (modulo ) is bounded or zero or unbounded as . Motivated further by -periodic point-counting result of Benedetto in arithmetic dynamics, we then also find that the average number of fixed points in -setting behaves in the same way as in -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 (-adic) integer polynomials, number fields and subfields of global function fields arising naturally in our polynomial discrete dynamical settings.

Paper Structure

This paper contains 15 sections, 52 theorems, 7 equations.

Table of Contents

  1. Introduction
  2. On Number of Integral Fixed Points of any Family of Polynomial Maps $\varphi_{p^{\ell},c}$
  3. The Number of $\mathbb{Z}_{p}\slash p\mathbb{Z}_{p}$-Fixed Points of any Family of Polynomial Maps $\varphi_{p^{\ell},c}$
  4. On Number of $\mathbb{Z}_{p}\slash p\mathbb{Z}_{p}$-Fixed Points of any Family of Polynomial Maps $\varphi_{(p-1)^{\ell},c}$
  5. The Number of $\mathbb{F}_{p}[t]\slash (\pi)$-Fixed Points of any Family of Polynomial Maps $\varphi_{p^{\ell},c}$
  6. Number of $\mathbb{F}_{p}[t]\slash (\pi)$-Fixed Points of any Family of Polynomial Maps $\varphi_{(p-1)^{\ell},c}$
  7. Average Number of $\mathcal{O}_{K}\slash p\mathcal{O}_{K}$-and $\mathbb{Z}_{p}\slash p\mathbb{Z}_{p}$-Fixed Points of any $\varphi_{p^{\ell},c}$ and $\varphi_{(p-1)^{\ell},c}$
  8. On Average Number of $\mathbb{F}_{p}[t]\slash (\pi)$-Fixed Points of any Family of $\varphi_{p^{\ell},c}$ & $\varphi_{(p-1)^{\ell},c}$
  9. On Density of Polynomials $\varphi_{p^{\ell},c}(x)$ with $N_{c}(p) = X_{c}(p) = p$ and $N_{c}(p) = X_{c}(p)=0$
  10. The Densities of Integer Polynomials $\varphi_{(p-1)^{\ell},c}(x)\in \mathbb{Z}_{p}[x]$ with $Y_{c}(p) = 1$ or $2$ or $0$
  11. On Local Densities of $f, g\in \mathbb{Z}_{p}[x]$ inducing Maximal orders in Associated Fields
  12. On the Number of Number fields $K_{f}\slash \mathbb{Q}$ with Bounded Absolute Discriminant
  13. On Fields $\mathbb{Q}_{f}\slash \mathbb{Q}$ with Bounded Absolute Discriminant and Prescribed Galois group
  14. On Fields $K_{f}\slash \mathbb{Q}$ with Bounded Absolute Discriminant and Prescribed Galois group
  15. On the Number of Intermediate fields $L$ of $H_{f_{c(t)}}\slash \mathbb{F}_{p}(t)$ and also $\Tilde{L}$ of $H_{g_{c(t)}}\slash \mathbb{F}_{p}(t)$

Key Result

Theorem 1.1

Let $K\slash \mathbb{Q}$ be any number field of degree $n \geq 2$ with the ring of integers $\mathcal{O}_{K}$, and in which any fixed prime integer $p\geq 3$ is inert. Let $\varphi_{p, c}$ be a polynomial map defined by $\varphi_{p, c}(z) = z^p + c$ for all $c, z\in\mathcal{O}_{K}$. Then the number

Theorems & Definitions (109)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Conjecture 1.6
  • Conjecture 1.7
  • Corollary 1.8
  • Conjecture 1.9
  • Conjecture 1.10
  • ...and 99 more