# Functional Decomposition using Principal Subfields

## TL;DR

This paper introduces a new algorithm leveraging principal subfields and subfield-intersection techniques to efficiently compute the subfield lattice of a rational function, improving the process of decomposing functions into simpler components.

## Contribution

It presents a novel Las Vegas algorithm that enhances complexity and runtime for identifying all non-equivalent complete decompositions of rational functions.

## Key findings

- Improved algorithmic complexity for subfield lattice computation
- Faster identification of rational function decompositions
- Enhanced practical performance in decomposition tasks

## Abstract

Let $f\in K(t)$ be a univariate rational function. It is well known that any non-trivial decomposition $g \circ h$, with $g,h\in K(t)$, corresponds to a non-trivial subfield $K(f(t))\subsetneq L \subsetneq K(t)$ and vice-versa. In this paper we use the idea of principal subfields and fast subfield-intersection techniques to compute the subfield lattice of $K(t)/K(f(t))$. This yields a Las Vegas type algorithm with improved complexity and better run times for finding all non-equivalent complete decompositions of $f$.