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

Brian Kintu

TL;DR

This work investigates the count and statistical behavior of integral fixed points and fixed orbits for polynomial maps of the forms φ_{p,c}(z)=z^p+c and φ_{p-1,c}(z)=z^{p-1}+c modulo a prime p, framing the problem within arithmetic statistics. Using elementary mod $p$ reductions of the fixed-point equations and good reduction arguments, it derives exact fixed-point counts: $N_c(p)\in\{3,0\}$ for φ_{p,c} and $M_c(p)\in\{1,2,0\}$ for φ_{p-1,c}, with the counts depending on residue classes of $c$ modulo $p$. The paper then analyzes averages over $c$ and densities of polynomials achieving these counts, proving that the average total fixed points converge to 3 and that the densities for the nontrivial counts are zero, while the case of no fixed points occurs with density 1. Extending to number-field questions, it applies Bhargava–Shankar–Wang results to bound the number of associated fields $K_f$ and $L_g$ with bounded discriminant and shows positive lower bounds for monogenic fields with symmetric Galois groups, highlighting a rich intersection of arithmetic dynamics, density theorems, and algebraic number theory.

Abstract

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.

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

TL;DR

This work investigates the count and statistical behavior of integral fixed points and fixed orbits for polynomial maps of the forms φ_{p,c}(z)=z^p+c and φ_{p-1,c}(z)=z^{p-1}+c modulo a prime p, framing the problem within arithmetic statistics. Using elementary mod reductions of the fixed-point equations and good reduction arguments, it derives exact fixed-point counts: for φ_{p,c} and for φ_{p-1,c}, with the counts depending on residue classes of modulo . The paper then analyzes averages over and densities of polynomials achieving these counts, proving that the average total fixed points converge to 3 and that the densities for the nontrivial counts are zero, while the case of no fixed points occurs with density 1. Extending to number-field questions, it applies Bhargava–Shankar–Wang results to bound the number of associated fields and with bounded discriminant and shows positive lower bounds for monogenic fields with symmetric Galois groups, highlighting a rich intersection of arithmetic dynamics, density theorems, and algebraic number theory.

Abstract

In this first article of a multi-part series, we inspect a surprising relationship between the set of fixed points of a polynomial map defined by for all and the coefficient , where 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 in arithmetic dynamics, we then first prove that for any prime , the average number of distinct integral fixed points of any modulo is or as tends to infinity. Inspired further by a conjecture of Hutz on rational periodic points of for any prime in arithmetic dynamics, we then also prove that the average number of distinct integral fixed points of any modulo 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 the irreducible integer polynomials and number fields arising naturally in our polynomial discrete dynamical settings.
Paper Structure (8 sections, 24 theorems, 3 equations)

This paper contains 8 sections, 24 theorems, 3 equations.

Table of Contents

  1. Introduction
  2. On the Number of Integral Fixed Points of any Family of Polynomial Maps $\varphi_{p,c}$
  3. The Number of Integral Fixed Points of any Family of Polynomial Maps $\varphi_{p-1,c}$
  4. On the Average Number of Fixed Points of any Polynomial Map $\varphi_{p,c}$ and $\varphi_{p-1,c}$
  5. On the Density of Monic Integer Polynomials $\varphi_{p,c}(x)$ with the number $N_{c}(p) = 3$
  6. On Densities of Monic Integer Polynomials $\varphi_{p-1,c}(x)$ with $M_{c}(p) = 1$ & $M_{c}(p) = 2$
  7. On Density of Integer Monics $\varphi_{p,c}(x)$ with $N_{c}(p) = 0$ and $\varphi_{p-1,c}(x)$ with $M_{c}(p) = 0$
  8. On the Number of Number fields $K_{f}$ & $L_{g}$ with Bounded Absolute Discriminant

Key Result

Theorem 1.1

Let $p\geq 3$ be any fixed prime integer, and assume Theorem theorem 3.2.1. Let $\varphi_{p, c}$ be a map defined by $\varphi_{p, c}(z) = z^p+c$ for all $c, z\in\mathbb{Z}$. The number of distinct integral fixed points of any $\varphi_{p,c}$ modulo $p$ is $3$ or zero.

Theorems & Definitions (54)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 1.3
  • Example 1.4
  • Example 1.5
  • Conjecture 1.6
  • Conjecture 1.7
  • Corollary 1.8
  • Conjecture 1.9
  • Theorem 1.10
  • ...and 44 more