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 $\alpha_1,\alpha_2,\beta\in L$. 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 $p$.