Post-critically finite polynomials in characteristic $p$

Abstract

It follows from the compactness of the connectedness locus that the locus of PCF polynomials in the moduli space of polynomials of degree d (at least 2) over the complex number is a countable union of zero-dimensional subvarieties. Over fields of characteristic p the analogous claim is certainly false, so it is natural to ask how large the components of the PCF locus can be. We show that the components of the PCF locus in the moduli space of polynomials of degree d over an algebraically closed field of characteristic p>0 have dimension at most d/p and that this bound is sharp, in the strong sense that for every p and d there are infinitely many distinct PCF components whose dimension is the greatest integer no greater than this bound (except in the case $p=d=2$).