Geometry of PCF parameters in spaces of quadratic polynomials

TL;DR

This paper extends the classification of algebraic relations among postcritically finite parameters from pairs to higher-dimensional subvarieties in complex space, providing uniform bounds and optimal estimates for such relations.

Contribution

It generalizes previous results to subvarieties of any dimension, establishing criteria for the finiteness of PCF parameter sets and computing optimal bounds for curves of any degree.

Findings

Characterization of algebraic relations among PCF parameters in higher dimensions

Uniform bounds on PCF pairs on non-special curves in a22

Optimal bounds for PCF parameters on degree d curves

Abstract

We study algebraic relations among postcritically finite (PCF) parameters in the family $f_c(z) = z^2 + c$. Ghioca, Krieger, Nguyen and Ye proved that an algebraic curve in $\mathbb{C}^2$ contains infinitely many PCF pairs $(c_1, c_2)$ if and only if the curve is special (i.e., the curve is a vertical or horizontal line through a PCF parameter, or the curve is the diagonal). Here we extend this result to subvarieties of $\mathbb{C}^n$ for any $n\geq 2$. Consequently, we obtain uniform bounds on the number of PCF pairs on non-special curves in $\mathbb{C}^2$ and the number of PCF parameters in real algebraic curves in $\mathbb{C}$, depending only on the degree of the curve. We also compute the optimal bound for the general curve of degree $d$.