Uniform unlikely intersections for unicritical polynomials

TL;DR

This paper establishes a uniform bound on the number of parameters for which two distinct points are simultaneously preperiodic under a family of unicritical polynomials, revealing a deep intersection property.

Contribution

It proves the existence of a universal constant C(d) bounding the number of such parameters for any pair of points with distinct d-th powers.

Findings

Existence of a uniform bound C(d) for preperiodic intersections.

Bound applies to all pairs with a^d ≠ b^d.

Results contribute to understanding unlikely intersections in polynomial families.

Abstract

Fix $d\geq2$ and let $f_{t}(z)=z^{d}+t$ be the family 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 $C(d)$.