The package provides Groebner bases computation interface in pure Julia with the performance comparable to Singular.
Groebner.jl
works over finite fields and the rationals, and supports various monomial orderings.
For documentation and more please check out https://sumiya11.github.io/Groebner.jl
Our package works with polynomials from AbstractAlgebra.jl
, DynamicPolynomials.jl
, and Nemo.jl
. We will demonstrate the usage on a simple example. Lets first create a ring of polynomials in 3 variables
using AbstractAlgebra
R, (x1, x2, x3) = PolynomialRing(QQ, ["x1", "x2", "x3"]);
Then we can define a simple polynomial system
polys = [
x1 + x2 + x3,
x1*x2 + x1*x3 + x2*x3,
x1*x2*x3 - 1
];
And compute the Groebner basis passing the system to groebner
using Groebner
G = groebner(polys)
3-element Vector{AbstractAlgebra.Generic.MPoly{Rational{BigInt}}}:
x1 + x2 + x3
x2^2 + x2*x3 + x3^2
x3^3 - 1
We compare the runtime of our implementation against the ones from Singular
and Maple
computer algebra systems. The table below lists measured runtimes of Groebner basis routine for several standard benchmark systems in seconds
System | Groebner.jl | Singular | Maple |
---|---|---|---|
cyclic-7 | 0.08 s | 1.4 s | 0.08 s |
cyclic-8 | 1.3 s | 40 s | 1.1 s |
katsura-10 | 0.8 s | 71 s | 0.9 s |
katsura-11 | 5.8 s | 774 s | 10 s |
eco-12 | 2.0 s | 334 s | 1.6 s |
eco-13 | 8.8 s | 5115 s | 13 s |
noon-7 | 0.1 s | 0.3 s | 0.15 s |
noon-8 | 1.0 s | 3.3 s | 1.1 s |
The bases are computed in degrevlex
monomial ordering over finite field of characteristic
We emphasize that Groebner.jl
is a specialized library while Singular
is an extensive general purpose computer algebra system.
This library is maintained by Alexander Demin ([email protected])
We would like to acknowledge Jérémy Berthomieu, Christian Eder and Mohab Safey El Din as this Library is inspired by their work "msolve: A Library for Solving Polynomial Systems". We are also grateful to Max-Planck-Institut für Informatik for assistance in producing benchmarks.
Special thanks goes to Vladimir Kuznetsov for providing the sources of his F4 implementation.
If you find Groebner.jl useful in your work, you can cite the following preprint
@misc{groebnerjl2023,
title = {Groebner.jl: A package for Gr\"obner bases computations in Julia},
author = {Alexander Demin and Shashi Gowda},
year = {2023},
eprint = {2304.06935},
url = {https://arxiv.org/abs/2304.06935}
}