Nearly Optimal Algorithms for the Decomposition of Multivariate Rational Functions and the Extended L\"uroth's Theorem

TL;DR

This paper introduces probabilistic and deterministic algorithms for decomposing multivariate rational functions and computing functions related to the extended L"uroth's Theorem, achieving near-optimal efficiency in certain cases.

Contribution

It presents new probabilistic and deterministic algorithms for function decomposition and extends the L"uroth's Theorem with polynomial-time solutions.

Findings

Probabilistic algorithms are nearly optimal for fixed n and large degree d.

An indecomposability test based on gcd and Newton's polytope is proposed.

A polynomial-time algorithm is achieved with minor modifications to existing methods.

Abstract

The extended L\"uroth's Theorem says that if the transcendence degree of $\KK(\mathsf{f}_1,\dots,\mathsf{f}_m)/\KK$ is 1 then there exists $f \in \KK(\underline{X})$ such that $\KK(\mathsf{f}_1,\dots,\mathsf{f}_m)$ is equal to $\KK(f)$. In this paper we show how to compute $f$ with a probabilistic algorithm. We also describe a probabilistic and a deterministic algorithm for the decomposition of multivariate rational functions. The probabilistic algorithms proposed in this paper are softly optimal when $n$ is fixed and $d$ tends to infinity. We also give an indecomposability test based on gcd computations and Newton's polytope. In the last section, we show that we get a polynomial time algorithm, with a minor modification in the exponential time decomposition algorithm proposed by Gutierez-Rubio-Sevilla in 2001.