Computing the fixing group of a rational function

TL;DR

This paper presents polynomial time algorithms for computing the fixing group of a rational function and the fixed field of a subgroup of automorphisms of K(x), advancing computational algebra methods.

Contribution

It introduces efficient algorithms for determining the fixing group of a rational function and the fixed field of subgroups of automorphisms in polynomial time.

Findings

Algorithms run in polynomial time

Successfully compute fixing groups for rational functions

Enhance computational tools in algebraic function field theory

Abstract

Let G=Aut_K (K(x)) be the Galois group of the transcendental degree one pure field extension K(x)/K. In this paper we describe polynomial time algorithms for computing the field Fix(H) fixed by a subgroup H < G and for computing the fixing group G_f of a rational function f in K(x).