Computing conjugating sets and automorphism groups of rational functions

TL;DR

This paper develops efficient algorithms to compute conjugating sets and automorphism groups of rational functions on the projective line, leveraging dynamical structures, with implementations demonstrating superior performance over naive methods.

Contribution

The paper introduces novel algorithms for computing automorphism groups and conjugating maps of rational functions, optimized for different dynamical contexts and implemented in Sage.

Findings

Algorithms outperform naive Groebner basis methods

Effective for fields like finite fields and rationals

Demonstrated on hundreds of random endomorphisms

Abstract

Let phi and psi be endomorphisms of the projective line of degree at least 2, defined over a noetherian commutative ring R with unity. From a dynamical perspective, a significant question is to determine whether phi and psi are conjugate (or to answer the related question of whether a given map phi has a nontrivial automorphism). We show that the space of automorphisms of P^1 conjugating phi to psi is a finite subscheme of PGL(2) (respectively that the automorphism group of phi is a finite group scheme). We construct efficient algorithms for computing the set of conjugating maps (resp. the group of automorphisms) when R is a field. Each of our algorithms takes advantage of different dynamical structures, so context (e.g., field of definition and degree of the map) determines the preferred algorithm. We have implemented them in Sage when R is a finite field or the field of rational numbers, and we give running times for computing automorphism groups for hundreds of random endomorphisms of P^1. These examples demonstrate the superiority of these new algorithms over a naive approach using Groebner bases.