Counting the number of ℤ_p-and \mathbbF_p[t]-fixed points of a discrete dynamical system with applications from arithmetic statistics, III

2025/05/30 by Brian Kintu, Kintu, Brian
Mathematics · #Dynamical Systems (math.DS) #FOS: Mathematics #Mathematical Dynamics and Fractals #Number Theory (math.NT) #advanced mathematical theories

paper · pdf · doi:10.48550/arxiv.2505.24565

openalex publication_date 2025/05/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this follow-up paper, we again inspect a surprising relationship between the set of fixed points of a polynomial map φ_{d, c} defined by φ_{d, c}(z) = z^d + c for all c, z ∈ O_K or ∈ ℤ_p or ∈ \mathbbF_p[t] and the coefficient c, where K is any number field of degree n > 1, p>2 is any prime, ℤ_p (resp., \mathbbF_p[t]) is the ring of all p-adic integers (resp., the ring of all polynomials over a finite field \mathbbF_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 ℚ_p-periodic points in arithmetic dynamics. In doing so, we then first prove that for any prime p≥ 3 and for any ℓ ∈ ℤ_{≥ 1}, the average number of distinct fixed points of any φ_{p^ℓ, c} modulo prime pO_K (modulo pℤ_p) is bounded or zero or unbounded as c→ ∞ . Motivated further by \mathbbF_p(t)-periodic point-counting result of Benedetto in arithmetic dynamics, we then also find that the average number of fixed points in \mathbbF_p[t]-setting behaves in the same way as in 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.

Citations

• Counting the number of integral fixed points of a discrete dynamical system with applications from arithmetic statistics, I
• An improvement on Schmidt's bound on the number of number fields of bounded discriminant and small degree
• Rational Periodic Points of x^d+c and Fermat-Catalan Equations
• Upper bounds on number fields of given degree and bounded discriminant
• Squarefree values of polynomial discriminants I
• Determination of all rational preperiodic points for morphisms of PN
• On Poonen's Conjecture Concerning Rational Preperiodic Points of Quadratic Maps
• The density of discriminants of quintic rings and fields
• Preperiodic points of polynomials over global fields
• Cycles of Quadratic Polynomials and Rational Points on a Genus-Two Curve

Related

• Counting the number of O_K-fixed points of a discrete dynamical system with applications from arithmetic statistics, II
• Counting the number of integral fixed points of a discrete dynamical system with applications from arithmetic statistics, I
• Counting the number of 2-periodic O_K-points of a discrete dynamical system with applications from arithmetic statistics, V
• Counting the number of n-periodic ℤ_p-and \mathbbF_p[t]-points of a discrete dynamical system with applications from arithmetic statistics, VI
• Counting the number of 2-periodic integral points of a discrete dynamical system with applications from arithmetic statistics, IV