Groebner.jl is a package for fast and generic Gröbner bases computations based on Faugère's F4 algorithm^{[1]} written in Julia.

To install Groebner.jl, run the following in the Julia REPL:

`using Pkg; Pkg.add("Groebner")`

See Interface for a description of all exported functions. For a quick introduction to Groebner bases we refer to Tutorials. Meanwhile, below are simple examples.

Currently, polynomials from AbstractAlgebra.jl, DynamicPolynomials.jl, and Nemo.jl are supported as input.

First, import AbstractAlgebra.jl. Then, we can create an array of polynomials over a finite field

```
using AbstractAlgebra
R, (x, y, z) = PolynomialRing(GF(2^31 - 1), ["x", "y", "z"])
polys = [x^2 + y + z, x*y + z];
```

and compute the Groebner basis with the `groebner`

command

```
using Groebner
basis = groebner(polys)
```

```
4-element Vector{AbstractAlgebra.Generic.MPoly{AbstractAlgebra.GFElem{Int64}}}:
y^3 + y^2*z + z^2
x*z + 2147483646*y^2 + 2147483646*y*z
x*y + z
x^2 + y + z
```

We can check if a set of polynomials forms a basis

`isgroebner(basis)`

`true`

Groebner.jl also provides several monomial orderings. For example, we can eliminate `z`

from the above system:

```
ordering = Lex(z) * DegRevLex(x, y) # z > x, y
groebner(polys, ordering=ordering)
```

```
2-element Vector{AbstractAlgebra.Generic.MPoly{AbstractAlgebra.GFElem{Int64}}}:
x^2 + 2147483646*x*y + y
x*y + z
```

You can find more information on monomial orderings in Groebner.jl in Monomial Orderings.

We will compute the basis of the `noon-2`

system^{[2]}

```
using DynamicPolynomials
@polyvar x1 x2
system = [10*x1*x2^2 - 11*x1 + 10,
10*x1^2*x2 - 11*x2 + 10]
groebner(system)
```

```
3-element Vector{DynamicPolynomials.Polynomial{DynamicPolynomials.Commutative{DynamicPolynomials.CreationOrder}, MultivariatePolynomials.Graded{MultivariatePolynomials.LexOrder}, Int64}}:
10x2 - 10x1 - 11x2² + 11x1²
110 - 121x2 - 100x2² + 100x1x2 + 110x2³
10 - 11x1 + 10x1x2²
```

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"^{[3]}. We are also grateful to The Max Planck Institute for Informatics and The MAX team at l'X for providing computational resources.

Special thanks goes to Vladimir Kuznetsov for providing the sources of his F4 implementation.

[1] | https://www-polsys.lip6.fr/~jcf/Papers/F99a.pdf |

[2] | https://www.jstor.org/stable/2101937 |

[3] | https://msolve.lip6.fr/ |

© Alexander Demin, Shashi Gowda, and others. Last modified: September 27, 2023. Website built with Franklin.jl and PkgPage.jl.