Geometry of PCF parameters in spaces of quadratic polynomials

2023/10/08 by DeMarco, Laura, Mavraki, Niki Myrto
#Dynamical Systems (math.DS) #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2310.05274

Abstract

We study algebraic relations among postcritically finite (PCF) parameters in the family f_c(z) = z² + c. Ghioca, Krieger, Nguyen and Ye proved that an algebraic curve in ℂ² contains infinitely many PCF pairs (c₁, c₂) 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 ℂⁿ for any n≥ 2. Consequently, we obtain uniform bounds on the number of PCF pairs on non-special curves in ℂ² and the number of PCF parameters in real algebraic curves in ℂ, depending only on the degree of the curve. We also compute the optimal bound for the general curve of degree d.

Related

• Special curves and postcritically-finite polynomials
• Classification of special curves in the space of cubic polynomials
• On quadratic curves over finite fields
• Bounded geometry for PCF-special subvarieties
• Cubic post-critically finite polynomials defined over ℚ