Fix $d\geq2$, 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 $a^{d}\neq b^{d}$, the number of $t\in\mathbb{C}$ such that $a$ and $b$ are both preperiodic for $f_{t}$ is at most $C(d)$.

• 37P05 - Arithmetic and non-Archimedean dynamical systems involving polynomial and rational maps
• 37P30 - Height functions; Green functions; invariant measures in arithmetic and non-Archimedean dynamical systems
• 37P50 - Dynamical systems on Berkovich spaces