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Uniform bounds on periodic points of polynomials with good reduction

Isaac Rajagopal, Robin Zhang

TL;DR

This work proves unconditional, explicit uniform bounds on the number of periodic points for a class of degree-$d$ polynomials with good reduction at a prime $\mathfrak{p}$ dividing $d$, showing $\#\mathrm{Per}_K(\phi) \le \mathrm{N}_{K/\mathbb{Q}}(\mathfrak{p}) \le d^D$ for number fields $K$ of degree $D$. The approach combines local $p$-adic dynamics—via a contraction/expansion framework and dynatomic polynomials—with a global transfer principle through embeddings into $p$-adic completions, yielding unconditional bounds that extend to unicritical polynomials $X^d+c$ with $c$ integral at a prime above $d$. The paper also derives refined exact-period bounds for prime-power degree using $m(\varpi,f,a_0,k)$ and dynatomic polynomials, and specializes to quadratic fields to classify $K$-rational periodic points of $\phi_{2,c}$, obtaining exact counts and confirming conjectures in specific fields. Overall, the results provide explicit, local-to-global uniform bounds and detailed structural consequences for periodic points in $p$-adic and quadratic settings, advancing the uniform boundedness program in arithmetic dynamics.

Abstract

We establish effective bounds on the number of periodic points of degree-$d$ polynomials $φ$ defined over $p$-adic fields and number fields, under a mild reduction hypothesis that is satisfied by all unicritical polynomials $X^d + c$ with $c$ integral at some prime dividing $d$. As a consequence, we verify the uniform boundedness conjecture for this class of polynomials over number fields $K$, giving the explicit uniform bound $\#\mathrm{Per}_K(φ) \leq d^{[K:\mathbb{Q}]}$.

Uniform bounds on periodic points of polynomials with good reduction

TL;DR

This work proves unconditional, explicit uniform bounds on the number of periodic points for a class of degree- polynomials with good reduction at a prime dividing , showing for number fields of degree . The approach combines local -adic dynamics—via a contraction/expansion framework and dynatomic polynomials—with a global transfer principle through embeddings into -adic completions, yielding unconditional bounds that extend to unicritical polynomials with integral at a prime above . The paper also derives refined exact-period bounds for prime-power degree using and dynatomic polynomials, and specializes to quadratic fields to classify -rational periodic points of , obtaining exact counts and confirming conjectures in specific fields. Overall, the results provide explicit, local-to-global uniform bounds and detailed structural consequences for periodic points in -adic and quadratic settings, advancing the uniform boundedness program in arithmetic dynamics.

Abstract

We establish effective bounds on the number of periodic points of degree- polynomials defined over -adic fields and number fields, under a mild reduction hypothesis that is satisfied by all unicritical polynomials with integral at some prime dividing . As a consequence, we verify the uniform boundedness conjecture for this class of polynomials over number fields , giving the explicit uniform bound .

Paper Structure

This paper contains 11 sections, 10 theorems, 17 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Uniform boundedness of periodic points
  3. Applications to exact periods
  4. Outline
  5. Periodic points in p-adic fields
  6. Bounds on the number of periodic points
  7. Bounds on exact periods in prime power degree
  8. Periodic points of quadratic polynomials
  9. From p-adic fields to number fields
  10. General number fields
  11. Quadratic number fields

Key Result

Theorem 3

Fix integers $d \geq 2$ and $D \geq 1$. Let $K/\mathbb{Q}$ be a number field of degree $D$, $\mathfrak{p}$ be a prime of $\mathcal{O}_K$ above a rational prime factor $p$ of $d$, and $\phi = \sum_{i=0}^d a_i X^i \in K[X]$ be a degree-$d$ polynomial with good reduction at $\mathfrak{p}$ such that $a_ which in particular is uniformly bounded above by $d^D$.

Figures (1)

  • Figure 1: In $F = \mathbb{Q}_2(\sqrt{5})$, the map $\phi(X) = X^2-1$ has two fixed points, $\alpha = \frac{1+\sqrt{5}}{2}$ and $\beta = \frac{1-\sqrt{5}}{2}$, and two period-$2$ points, $0$ and $1$. Their nearby points are attracted, while faraway points are repelled. The largest circle represents $\mathcal{O}_F = \mathbb{Z}_2[\alpha]$. The four medium-size circles represent open balls of radius $1$. The sixteen smaller circles represent open balls of radius $\frac{1}{2}$.

Theorems & Definitions (27)

  • Definition 1
  • Conjecture 2: Uniform boundedness conjecture for periodic points
  • Theorem 3
  • Remark 4
  • Theorem 5
  • Example 6: see Corollary \ref{['cor:2-general']}
  • Corollary 7
  • Proposition 2.1
  • Lemma 2.2
  • proof
  • ...and 17 more