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

Abstract

Let $L$ be a field of positive characteristic $p$ with a fixed algebraic closure $\overline{L}$, and let $α_1,α_2,β\in L$. For an integer $d\ge 2$, we consider the family of polynomials $f_λ(z) := z^d+λ$, parameterized by $λ\in\overline{L}$. Define $C(α_1,α_2;β)$ to be the set of all $λ\in\overline{L}$ for which there exist $m,n\in\mathbb{N}$ such that $f_λ^m(α_1)=f_λ^n(α_2)=β$. In other words, $C(α_1,α_2;β)$ consists of all $λ\in\overline{L}$ with the property that the orbit of $α_1$ collides with the orbit of $α_2$ under the same polynomial $f_λ$ precisely at the point $β$. Assuming $α_1,α_2,β$ are not all contained in a finite subfield of $L$, we provide explicit necessary and sufficient conditions under which $C(α_1,α_2;β)$ is infinite. We also discuss the remaining case where $α_1,α_2,β\in \overline{\mathbb F}_p$ 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 $p$.