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 $p$.

38 pages