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

Brian Kintu

TL;DR

The paper develops a detailed arithmetic-dynamics framework for counting $\mathcal{O}_{K}$-fixed points of polynomial maps $\varphi_{d,c}(z)=z^{d}+c$ and analyzes how these counts behave modulo inert primes. By combining finite-field reductions, good reduction, and density- and field-count techniques from arithmetic statistics, it proves precise fixed-point counts for families $\varphi_{p^{\ell},c}$ and $\varphi_{(p-1)^{\ell},c}$, along with their average and density behavior as the parameter $c$ grows. It then connects these dynamical counts to the arithmetic of number fields, obtaining density results for when no integral fixed points occur and deriving upper and lower bounds on the number of fields with bounded discriminant that arise from these dynamical systems, including monogenic fields with prescribed Galois groups. The work thus links discrete dynamical counting problems to broad arithmetic-statistical phenomena, providing universal average counts (e.g., $3$ or $0$ for certain families) and highlighting notable densities (such as $\zeta(2)^{-1}$ for monogenic fields) that illuminate the distribution of number fields generated by dynamical polynomials.

Abstract

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.

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

TL;DR

The paper develops a detailed arithmetic-dynamics framework for counting -fixed points of polynomial maps and analyzes how these counts behave modulo inert primes. By combining finite-field reductions, good reduction, and density- and field-count techniques from arithmetic statistics, it proves precise fixed-point counts for families and , along with their average and density behavior as the parameter grows. It then connects these dynamical counts to the arithmetic of number fields, obtaining density results for when no integral fixed points occur and deriving upper and lower bounds on the number of fields with bounded discriminant that arise from these dynamical systems, including monogenic fields with prescribed Galois groups. The work thus links discrete dynamical counting problems to broad arithmetic-statistical phenomena, providing universal average counts (e.g., or for certain families) and highlighting notable densities (such as for monogenic fields) that illuminate the distribution of number fields generated by dynamical polynomials.

Abstract

In this follow-up paper, we again inspect a surprising connection between the set of fixed points of a polynomial map defined by for all and the coefficient , where is any number field of degree and 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 -rational periodic points of any odd degree map in arithmetic dynamics. In doing so, we then first prove that for any real algebraic number field of degree , and for any prime and integer , the average number of distinct integral fixed points of any modulo prime ideal is or as . Motivated further by -rational periodic point-counting result of Benedetto on any for any prime and integer in arithmetic dynamics, we then also prove unconditionally that for any number field (not necessarily real) of degree , the average number of distinct integral fixed points of any modulo prime is or or as . 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.

Paper Structure

This paper contains 10 sections, 32 theorems, 6 equations.

Table of Contents

  1. Introduction
  2. On the Number of Integral Fixed Points of any Family of Polynomial Maps $\varphi_{p^{\ell},c}$
  3. The Number of Integral Fixed Points of any Family of Polynomial Maps $\varphi_{(p-1)^{\ell},c}$
  4. On the Average Number of Fixed Points of any Polynomial Map $\varphi_{p^{\ell},c}$ & $\varphi_{(p-1)^{\ell},c}$
  5. On the Density of Integer Polynomials $\varphi_{p^{\ell},c}(x)\in \mathcal{O}_{K}[x]$ with the Number $N_{c}(p) = 3$
  6. On Densities of Monic Integer Polynomials $\varphi_{(p-1)^{\ell},c}(x)\in \mathcal{O}_{K}[x]$ with $M_{c}(p) = 1$ or $2$
  7. The Density of Integer Monics $\varphi_{p^{\ell},c}(x)$ with $N_{c}(p) = 0$ & $\varphi_{(p-1)^{\ell},c}(x)$ with $M_{c}(p) = 0$
  8. On the Number of Algebraic Number fields with Bounded Absolute Discriminant
  9. On Fields $\mathbb{Q}_{f}$ and $\mathbb{Q}_{g}$ with Bounded Absolute Discriminant and Prescribed Galois group
  10. On Fields $K_{f}$ and $L_{g}$ with Bounded Absolute Discriminant and Prescribed Galois group

Key Result

Theorem 1.1

Let $K\slash \mathbb{Q}$ be any real number field of degree $n \geq 2$ with the ring of integers $\mathcal{O}_{K}$, and in which any fixed prime $p\geq 3$ is inert. Assume Thm theorem 3.2.1, and let $\varphi_{p, c}$ be a polynomial map defined by $\varphi_{p, c}(z) = z^p + c$ for all $c, z\in\mathca

Theorems & Definitions (68)

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