Integrality and Thurston Rigidity for Bicritical PCF Polynomials

TL;DR

This paper proves that bicritical PCF polynomials with periodic critical points have integral solutions at specific primes, using algebraic methods to support Thurston rigidity and transversality results.

Contribution

It provides an algebraic proof of Thurston rigidity implications for bicritical PCF polynomials and identifies primes where solutions are integral, expanding understanding of polynomial dynamics.

Findings

PCF solutions are integral at index divisor free primes

Existence of such primes is proven in all but finitely many cases

Primes are used to establish transversality in polynomial families

Abstract

We give an algebraic proof of an important consequence of Thurston rigidity for bicritical PCF polynomials with periodic critical points under certain mild assumptions. The key result is that when the family of bicritical polynomials is parametrized using dynamical Belyi polynomials, the PCF solutions are integral at certain special primes, which we term ``index divisor free primes.'' We prove the existence of index divisor free primes in all but finitely many cases and conjecture the complete list of exceptions. These primes are then used to prove transversality.