标题： 有限特征域上定义的多项式族轨道的碰撞 显示英文标题

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

摘要： 设$L$为一个正特征$p$的域，并固定其代数闭包$\overline{L}$，并让$\alpha_1,\alpha_2,\beta\in L$。 对于整数$d\ge 2$，我们考虑由$\lambda\in\overline{L}$参数化的多项式族$f_{\lambda}(z) := z^d+\lambda$。 定义$C(\alpha_1,\alpha_2;\beta)$为所有$\lambda\in\overline{L}$的集合，其中存在$m,n\in\mathbb{N}$使得$f_{\lambda}^m(\alpha_1)=f_{\lambda}^n(\alpha_2)=\beta$。 换句话说， $C(\alpha_1,\alpha_2;\beta)$包含所有$\lambda\in\overline{L}$，其特性是$\alpha_1$的轨道在相同的多项式$f_{\lambda}$下恰好在点$\beta$与$\alpha_2$的轨道相撞。 假设$\alpha_1,\alpha_2,\beta$不全包含在$L$的有限子域中，我们提供了明确的充要条件，使得$C(\alpha_1,\alpha_2;\beta)$为无限。 我们还讨论了剩余的情况，即$\alpha_1,\alpha_2,\beta\in \overline{\mathbb F}_p$，并提供了丰富的计算数据，这些数据暗示了一个有些令人惊讶的猜想。 我们的问题属于算术动力学中不寻常交点相关的一系列问题，这些问题主要在特征为$0$的域上进行研究。 在特征为$p$的情况下工作会带来显著的困难，但同时也揭示了我们问题的微妙之处，特别是在一些点位于有限域中或当$d$是$p$的幂时。 显示英文摘要

摘要： 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 $p$.