The package provides Groebner bases computation interface in pure Julia with the performance comparable to Singular.
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
julia> using AbstractAlgebra
julia> R, (x1, x2, x3) = PolynomialRing(QQ, ["x1", "x2", "x3"]);Then we can define a simple polynomial system
julia> polys = [
x1 + x2 + x3,
x1*x2 + x1*x3 + x2*x3,
x1*x2*x3 - 1
];And compute the Groebner basis passing the system to groebner
julia> using Groebner
julia> G = groebner(polys)
3-element Vector{AbstractAlgebra.Generic.MPoly{Rational{BigInt}}}:
x1 + x2 + x3
x2^2 + x2*x3 + x3^2
x3^3 - 1We 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.
If you discover a system where our package shows bad performance, you are very welcome to submit an issue!
This library is maintained by Alexander Demin (asdemin_2@edu.hse.ru)
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.