Rational Transformations and Invariant Polynomials

TL;DR

This paper investigates the factorization of rational transformations and invariant polynomials under finite group actions, extending known results and providing new insights into the structure and number of irreducible factors over various fields.

Contribution

It establishes a connection between the factorization of rational transformations and group actions on polynomials, extending previous results to non-cyclic groups and arbitrary fields.

Findings

Factorization relates to a known group action of G on monic polynomials.

Extended Lucas Reis's result to G-invariant irreducible polynomials.

Provided new bounds on the number of irreducible factors for non-cyclic G.

Abstract

Rational transformations of polynomials are extensively studied in the context of finite fields, especially for the construction of irreducible polynomials. In this paper, we consider the factorization of rational transformations with (normalized) generators of the field $K(x)^G$ of $G$-invariant rational functions for $G$ a finite subgroup of $\operatorname{PGL}_2(K)$, where $K$ is an arbitrary field. Our main theorem shows that the factorization is related to a well-known group action of $G$ on a subset of monic polynomials. With this, we are able to extend a result by Lucas Reis for $G$-invariant irreducible polynomials. Additionally, some new results about the number of irreducible factors of rational transformations for $Q$ a generator of $\mathbb{F}_q(x)^G$ are given when $G$ is non-cyclic.