Unlikely Intersection For Two-Parameter Families of Polynomials

TL;DR

This paper proves that for three distinct complex points, the set of quadratic polynomials with parameters making all three points preperiodic is not densely distributed in the parameter space.

Contribution

It establishes a non-density result for preperiodic points in two-parameter polynomial families, extending understanding of polynomial dynamics.

Findings

The set of parameters where three points are preperiodic is not Zariski dense.

Preperiodicity conditions impose strong algebraic restrictions.

Results apply to polynomials of degree at least 3 with complex parameters.

Abstract

Let $c_1, c_2, c_3$ be distinct complex numbers, and let $d\ge 3$ be an integer. We show that the set of all pairs $(a,b)\in \mathbb{C}\times \mathbb{C}$ such that each $c_i$ is preperiodic for the action of the polynomial $x^d+ax+b$ is not Zariski dense in the affine plane.