Collision of orbits for families of polynomials defined over fields of positive characteristic

TL;DR

This paper investigates when orbits of points under polynomial families over fields of positive characteristic collide, providing explicit conditions for infinite collision sets and exploring special cases with computational evidence.

Contribution

It offers explicit criteria for orbit collisions in positive characteristic, extending unlikely intersection problems into a new arithmetic dynamical setting.

Findings

Criteria for when $C( ext{alpha}_1, ext{alpha}_2; ext{beta})$ is infinite

Analysis of cases where points lie in finite fields

Computational data supporting a new conjecture

Abstract

Let $L$ be a field of positive characteristic $p$ with a fixed algebraic closure $\overline{L}$, and let $\alpha_1,\alpha_2,\beta\in L$. For an integer $d\ge 2$, we consider the family of polynomials $f_{\lambda}(z) := z^d+\lambda$, parameterized by $\lambda\in\overline{L}$. Define $C(\alpha_1,\alpha_2;\beta)$ to be the set of all $\lambda\in\overline{L}$ for which there exist $m,n\in\mathbb{N}$ such that $f_{\lambda}^m(\alpha_1)=f_{\lambda}^n(\alpha_2)=\beta$. In other words, $C(\alpha_1,\alpha_2;\beta)$ consists of all $\lambda\in\overline{L}$ with the property that the orbit of $\alpha_1$ collides with the orbit of $\alpha_2$ under the same polynomial $f_{\lambda}$ precisely at the point $\beta$. Assuming $\alpha_1,\alpha_2,\beta$ are not all contained in a finite subfield of $L$, we provide explicit necessary and sufficient conditions under which $C(\alpha_1,\alpha_2;\beta)$ is infinite. We also discuss the remaining case where $\alpha_1,\alpha_2,\beta\in \overline{\mathbb F}_p$ and provide ample computational data that suggest a somewhat surprising conjecture. Our problem fits into a long series of questions in the area of unlikely intersections in arithmetic dynamics, which have been primarily studied over fields of characteristic $0$. Working in characteristic $p$ adds significant difficulties, but also reveals the subtlety of our problem, especially when some of the points lie in a finite field or when $d$ is a power of $p$.