Geometry of PCF parameters in spaces of quadratic polynomials

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 $d$.