New families satisfying the Dynamical Uniform Boundedness Principle over function fields

TL;DR

This paper extends techniques to prove the Strong Uniform Boundedness Principle for new families of polynomials and rational functions over function fields, including quadratic polynomials in even characteristic.

Contribution

It introduces new families of functions satisfying the SUBP, expanding the scope of algebraic dynamics over function fields in positive characteristic.

Findings

Many new 1-parameter polynomial families satisfy the SUBP

Quadratic polynomials in even characteristic satisfy the SUBP

A new family of non-polynomial, non-Lattès rational functions satisfies the SUBP

Abstract

We extend a technique, originally due to the first author and Poonen, for proving cases of the Strong Uniform Boundedness Principle (SUBP) in algebraic dynamics over function fields of positive characteristic. The original method applied to unicritical polynomials for which the characteristic does not divide the degree. We show that many new 1-parameter families of polynomials satisfy the SUBP, including the family of all quadratic polynomials in even characteristic. We also give a new family of non-polynomial, non-Latt\`es rational functions that satisfies the SUBP.