Rational Transformations and Invariant Polynomials

2023/06/23 by Max Schulz, Schulz, Max
Computer Science · Mathematics · #Advanced Differential Equations and Dynamical Systems #Coding theory and cryptography #FOS: Mathematics #Number Theory (math.NT) #semigroups and automata theory

paper · pdf · doi:10.48550/arxiv.2306.13502

openalex publication_date 2023/06/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 PGL₂(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 \mathbbF_q(x)^G are given when G is non-cyclic.

Related

• Invariant Rational Functions, Linear Fractional Transformations and Irreducible Polynomials over Finite Fields
• On the existence and number of invariant polynomials
• Invariant theory of a special group action on irreducible polynomials over finite fields
• Generic orbits, normal bases, and generation degree for fields of rational invariants
• Construction of irreducible polynomials through rational transformations