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Simple generators of rational function fields

Alexander Demin, Gleb Pogudin

TL;DR

This work addresses the problem of producing simple, interpretable generators for subfields of the rational function field $k(\mathbf{x})$ by leveraging OMS ideals and an evaluation–interpolation paradigm. It introduces a main algorithm that uses partial Gröbner basis information, sparse interpolation, and randomized subfield membership tests to incrementally build a compact generating set, avoiding full intermediate expression swell. The authors implement the method in Julia, integrate modular computations, and demonstrate substantial improvements in efficiency and generator quality across benchmarks and several application domains, notably structural identifiability and invariants. The results show that many subfields can be generated by a small number of low-degree, often polynomial, elements, with practical impact on model analysis and invariant theory.

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 Gröbner basis computation via sparse interpolation and efficient search for polynomials of a fixed degree in a subfield of the rational function field.

Simple generators of rational function fields

TL;DR

This work addresses the problem of producing simple, interpretable generators for subfields of the rational function field by leveraging OMS ideals and an evaluation–interpolation paradigm. It introduces a main algorithm that uses partial Gröbner basis information, sparse interpolation, and randomized subfield membership tests to incrementally build a compact generating set, avoiding full intermediate expression swell. The authors implement the method in Julia, integrate modular computations, and demonstrate substantial improvements in efficiency and generator quality across benchmarks and several application domains, notably structural identifiability and invariants. The results show that many subfields can be generated by a small number of low-degree, often polynomial, elements, with practical impact on model analysis and invariant theory.

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 Gröbner basis computation via sparse interpolation and efficient search for polynomials of a fixed degree in a subfield of the rational function field.
Paper Structure (33 sections, 16 theorems, 75 equations, 5 tables, 8 algorithms)

This paper contains 33 sections, 16 theorems, 75 equations, 5 tables, 8 algorithms.

Table of Contents

  1. Introduction
  2. Acknowledgements.
  3. Preliminaries
  4. Notation
  5. DeMillo-Lipton-Schwartz-Zippel lemma
  6. Univariate rational function interpolation
  7. Sparse polynomial interpolation
  8. Sparse rational function interpolation
  9. The degree of a blackbox function
  10. OMS Ideals
  11. Randomized subfield membership testing
  12. Building blocks of the algorithm
  13. Basic minimization
  14. Low-degree coefficients of Gröbner bases over rational function fields
  15. Finding polynomial generators
  16. ...and 18 more sections

Key Result

Lemma 2.1

Let $f \in k[\mathbf{x}]$ be a nonzero polynomial of degree $d$. The probability that $f(\bm{\gamma}) = 0$ is at most $d/|S|$.

Theorems & Definitions (71)

  • Example 1.1
  • Lemma 2.1: DeMillo-Lipton-Schwarz-Zippel demillo-liptonSchwartzzippel-lemma
  • Corollary 2.2
  • proof
  • Remark 2.3
  • Remark 2.4
  • Remark 2.5
  • Remark 2.6: Diversification diversification
  • Remark 2.7
  • Proposition 2.8
  • ...and 61 more