Uniform unlikely intersections for unicritical polynomials

Abstract

Fix $d\ge 2$, and consider the family $f_t(z)=z^d+t$ of polynomials parameterized by $t\in \mathbb{C}$. In this article, we will show that there exists a constant $C(d)$ such that for any $a,b\in \mathbb{C}$ with $a^d\ne b^d$, the number of $t\in \mathbb{C}$ such that $a$ and $b$ are both preperiodic for $f_t$ is at most $C(d)$.

References in corpus (1)

• A metric of mutual energy and unlikely intersections for dynamical systems