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The GroebnerWalk.jl package for OSCAR

Kamillo Ferry, Francesco Nowell

TL;DR

This work introduces GroebnerWalk.jl, an implementation of the Gröbner walk that leverages Gröbner fan geometry to transform a Gröbner basis from a start ordering to a target ordering. It implements two variants, the standard walk and the generic walk, and provides a unified interface within OSCAR with diagnostic tracing capabilities. Through experiments on zero-dimensional lex bases over $\mathbb{Q}$ and $\mathbb{F}_p$ and higher-dimensional elimination problems, the results show that the standard walk can be competitive with classical Buchberger-based methods, while the generic walk may underperform due to a naive normal-form computation and backend weight limitations. The work outlines practical improvements and future directions, including optimized normal-form computation, reduced reliance on the Singular backend by OSCAR-native implementations, and starter-order detection to reduce initial basis computation.

Abstract

Computing Gröbner bases is known to have a very high upper bound on computation time with respect to input length. Due to the connection between polyhedral geometry and Gröbner bases through the Gröbner fan, one can attempt an incremental approach to compute Gröbner bases. First computing a Gröbner basis with respect to an `easy' term order and transforming that result to a Gröbner basis with respect to the desired term order by using information about this polyhedral fan is done by a family of algorithms termed as Gröbner walk. We implemented two variants of the Gröbner walk in the computer algebra system OSCAR and compared their performance with classical Gröbner basis methods already found in OSCAR.

The GroebnerWalk.jl package for OSCAR

TL;DR

This work introduces GroebnerWalk.jl, an implementation of the Gröbner walk that leverages Gröbner fan geometry to transform a Gröbner basis from a start ordering to a target ordering. It implements two variants, the standard walk and the generic walk, and provides a unified interface within OSCAR with diagnostic tracing capabilities. Through experiments on zero-dimensional lex bases over and and higher-dimensional elimination problems, the results show that the standard walk can be competitive with classical Buchberger-based methods, while the generic walk may underperform due to a naive normal-form computation and backend weight limitations. The work outlines practical improvements and future directions, including optimized normal-form computation, reduced reliance on the Singular backend by OSCAR-native implementations, and starter-order detection to reduce initial basis computation.

Abstract

Computing Gröbner bases is known to have a very high upper bound on computation time with respect to input length. Due to the connection between polyhedral geometry and Gröbner bases through the Gröbner fan, one can attempt an incremental approach to compute Gröbner bases. First computing a Gröbner basis with respect to an `easy' term order and transforming that result to a Gröbner basis with respect to the desired term order by using information about this polyhedral fan is done by a family of algorithms termed as Gröbner walk. We implemented two variants of the Gröbner walk in the computer algebra system OSCAR and compared their performance with classical Gröbner basis methods already found in OSCAR.

Paper Structure

This paper contains 6 sections, 3 theorems, 12 equations, 1 figure, 3 tables.

Table of Contents

  1. Introduction
  2. Basics of Gröbner fans
  3. Functionality
  4. Technical Contribution
  5. Comparison to classical Gröbner basis algorithms
  6. Future directions

Key Result

Theorem 2

For a non-zero ideal $I \triangleleft R$, the following sets are in one-to-one correspondence:

Figures (1)

  • Figure 1: The Gröbner fan of the ideal $\langle y^4+ x^3-x^2+x,x^4\rangle$. Each point denotes an intermediate weight vector for which a Gröbner basis is computed. For implementation reasons, we choose an integer weight vector in every intermediate step.

Theorems & Definitions (5)

  • Example 1
  • Theorem 2
  • Lemma 3
  • Lemma 4
  • Example 5