On simultaneously preperiodic points for one-parameter families of polynomials in characteristic $p$

Abstract

For a field $L$ of characteristic $p$, a polynomial $f \in \overline{\mathbb{F}}_p[x]$ and $\alpha, \beta \in L$, let $\mathrm{Prep}(f;\alpha,\beta)$ be the set of all $\lambda \in \overline{L}$ such that both $\alpha$ and $\beta$ are preperiodic under the action of $f_{\lambda}(x) := f(x) + \lambda$. Ghioca and Hsia proved that for certain families of polynomials, this set is infinite if and only if $f(\alpha)=f(\beta)$ or $\alpha, \beta \in \overline{\mathbb{F}}_p$. Building on their work, we determine when $\mathrm{Prep}(f;\alpha,\beta)$ is infinite for most of the remaining binomial cases that were left open. Specifically, let $f(x)=c_1 x^{d_1} + c_2 x^{d_2} \in \overline{\mathbb{F}}_p[x]$, where $c_i \in \overline{\mathbb{F}}_p^*$, $1 \le d_1 < d_2$ and $d_i=p^{\ell_i}s_i$ with $\ell_i \ge 0$ and $p \nmid s_i$. We prove that if $p^{\ell_2}(s_2-1) < p^{\ell_1}(s_1-1)$, then $\mathrm{Prep}(f;\alpha,\beta)$ is infinite if and only if $f(\alpha)=f(\beta)$ or $\alpha, \beta \in \overline{\mathbb{F}}_p$. The key idea of the proof is to use the parameters $\lambda_{\overline{\alpha}} := \overline{\alpha} - f(\overline{\alpha})$ associated to suitable elements $\overline{\alpha} \in \overline{L}$ satisfying $f(\overline{\alpha})=f(\alpha)$. As an application, we extend the work of Asgarli and Ghioca on the colliding orbits problem to binomials satisfying $s_2>1$ and $p^{\ell_2}(s_2-1) < p^{\ell_1}(s_1-1)$.