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Groebner.jl: A package for Gröbner bases computations in Julia

Alexander Demin, Shashi Gowda

TL;DR

Groebner.jl delivers a native Julia implementation of the $F4$ algorithm for Gröbner bases, supporting computations over $\\bZ/p\\bbZ$ and $\\bbQ$ with a focus on the degree reverse lexicographical order. It combines a distributed polynomial representation, packed monomial encoding, and multiple arithmetic strategies, and leverages multi-modular computation with tracing and batched application to accelerate rational reconstructions via CRT. The framework integrates with existing Julia symbolic tools (e.g., AbstractAlgebra, DynamicPolynomials, Symbolics.jl) and demonstrates competitive performance against established systems, aided by parallelism and batched linear algebra. These contributions enhance the efficiency, portability, and extensibility of Gröbner-bases computations in Julia, enabling scalable use in research and applications across symbolic computation and dynamical systems modeling.

Abstract

We present Groebner.jl, a Julia package for computing Groebner bases with the F4 algorithm. Groebner.jl is an efficient, portable, and open-source software. Groebner.jl works over integers modulo a prime and over the rationals, supports basic multi-threading, and specializes in computation in the degree reverse lexicographical monomial ordering. The implementation incorporates various symbolic computation techniques and leverages the Julia type system and tooling, which allows Groebner.jl to compete with the existing state of the art, in many instances outperform it, and exceed them in extensibility. Groebner.jl is freely available at https://github.com/sumiya11/Groebner.jl.

Groebner.jl: A package for Gröbner bases computations in Julia

TL;DR

Groebner.jl delivers a native Julia implementation of the algorithm for Gröbner bases, supporting computations over and with a focus on the degree reverse lexicographical order. It combines a distributed polynomial representation, packed monomial encoding, and multiple arithmetic strategies, and leverages multi-modular computation with tracing and batched application to accelerate rational reconstructions via CRT. The framework integrates with existing Julia symbolic tools (e.g., AbstractAlgebra, DynamicPolynomials, Symbolics.jl) and demonstrates competitive performance against established systems, aided by parallelism and batched linear algebra. These contributions enhance the efficiency, portability, and extensibility of Gröbner-bases computations in Julia, enabling scalable use in research and applications across symbolic computation and dynamical systems modeling.

Abstract

We present Groebner.jl, a Julia package for computing Groebner bases with the F4 algorithm. Groebner.jl is an efficient, portable, and open-source software. Groebner.jl works over integers modulo a prime and over the rationals, supports basic multi-threading, and specializes in computation in the degree reverse lexicographical monomial ordering. The implementation incorporates various symbolic computation techniques and leverages the Julia type system and tooling, which allows Groebner.jl to compete with the existing state of the art, in many instances outperform it, and exceed them in extensibility. Groebner.jl is freely available at https://github.com/sumiya11/Groebner.jl.
Paper Structure (13 sections, 1 theorem, 5 equations, 2 figures, 5 tables, 1 algorithm)

This paper contains 13 sections, 1 theorem, 5 equations, 2 figures, 5 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Background
  3. Notations and definitions
  4. F4 algorithm
  5. Functionality
  6. The implementation
  7. Polynomial representation
  8. Linear algebra
  9. Multi-modular computation
  10. Learn and apply in batches
  11. The use of multi-threading
  12. Benchmarks
  13. Concluding remarks

Key Result

theorem 1

Let $I = \langle G \rangle$ be the ideal generated by a finite set of polynomials $G \subset \mathcal{R}$. Then, $G$ is a Gröbner basis of $I$ with respect to monomial ordering $<$ if and only if the S-polynomial of $g_i, g_j$ for all $g_i, g_j \in G$ reduces to zero by $G$.

Figures (2)

  • Figure 1: Monomial division mask. A monomial $x^2yz^3$ is represented by an array of integers $\{2, 1, 3\}$. The array, in turn, is compressed to a 16-bit machine integer, 5 bit per exponent.
  • Figure 2: Macaulay matrix in Groebner.jl: (a) initially, after symbolic preprocessing (b) after reducing each row in the block $C|D$ with $A|B$ (c) after inter-reducing the rows of $D'$.

Theorems & Definitions (3)

  • definition 1
  • theorem 1
  • definition 2