Simple generators of rational function fields

TL;DR

This paper introduces an efficient algorithm for simplifying generators of rational function fields, improving computational performance and result quality, with practical applications demonstrated across various domains.

Contribution

It presents a novel algorithm that simplifies generators of rational function fields using partial Gr"obner basis computation and sparse interpolation, enhancing efficiency and results.

Findings

Algorithm outperforms existing methods in efficiency

Produces simpler, more manageable generators

Demonstrated utility in structural parameter identifiability

Abstract

Consider a subfield of the field of rational functions in several indeterminates. We present an algorithm that, given a set of generators of such a subfield, finds a simple generating set. We provide an implementation of the algorithm and show that it improves upon the state of the art both in efficiency and the quality of the results. Furthermore, we demonstrate the utility of simplified generators through several case studies from different application domains, such as structural parameter identifiability. The main algorithmic novelties include performing only partial Gr\"obner basis computation via sparse interpolation and efficient search for polynomials of a fixed degree in a subfield of the rational function field.