Simultaneously preperiodic points for families of polynomials in characteristic $p$
Abstract
The first author proved that given a field $K$ of characteristic $p \gt 0$, given an integer $d\ge 2$, and also given $\alpha ,\beta \in K$, for the family of polynomials $f_\lambda (x):=x^d+\lambda $ (parameterized by $\lambda \in \overline{K}$) there exist infinitely many $\lambda \in \overline{K}$ such that both $\alpha $ and $\beta $ are preperiodic under the action of $f_\lambda $ if and only if at least one of the following three conditions holds: (i) $\alpha ^d=\beta ^d$, (ii) $\alpha ,\beta \in \overline{\mathbb F}_p$, and (iii) $d=p^\ell $ for some $\ell \in \mathbb {N}$ and $\alpha -\beta \in \overline{\mathbb F}_p$. In the present paper, we generalize these results by proving that for any $f\in \overline{\mathbb F}_p[x]$ of degree $d\ge 2$ satisfying a mild condition (which is already satisfied when $p\nmid d$), and for any $\alpha ,\beta \in K$, there exist infinitely many $\lambda \in \overline{K}$ such that both $\alpha $ and $\beta $ are preperiodic for the polynomials $f_\lambda (x):=f(x)+\lambda $ if and only if at least one of the following two conditions holds: (i) $f(\alpha )=f(\beta )$, (ii) $\alpha ,\beta \in \overline{\mathbb F}_p$.
