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 paper investigates fixed points of polynomial maps over various rings, revealing their statistical behavior and applying these findings to count related algebraic structures in arithmetic dynamics.

Contribution

It establishes bounds and asymptotic behaviors for fixed points of polynomial maps over rings like _p[t] and _p, extending previous results and connecting to arithmetic statistics.

Findings

Average fixed points are bounded, zero, or unbounded as parameters grow.

Fixed point counts behave similarly over _p[t] and _p.

Results lead to counting algebraic structures like number fields and subfields.

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.