A finiteness result for post-critically finite polynomials

TL;DR

This paper proves that the set of post-critically finite polynomials of a fixed degree with algebraic coefficients of bounded degree is finite and effectively computable, using height relations in moduli space.

Contribution

It establishes a finiteness result for post-critically finite polynomials with bounded algebraic degree, and explicitly computes this set for degree 3 and algebraic degree 1.

Findings

Set of such polynomials is finite and effectively computable.

Explicit computation for degree 3, algebraic degree 1 case.

Relation between naive height and Silverman's critical height.

Abstract

We show that the set of complex points in the moduli space of polynomials of degree d corresponding to post-critically finite polynomials is a set of algebraic points of bounded height. It follows that for any B, the set of conjugacy classes of post-critically finite polynomials of degree d with coefficients of algebraic degree at most B is a finite and effectively computable set. In the case d=3 and B=1 we perform this computation. The proof of the main result comes down to finding a relation between the "naive" height on the moduli space, and Silverman's critical height.