Gröbner walk

GroebnerWalk.jl is a Julia package providing implementations of Gröbner walk algorithms for computing Gröbner bases over fields on top of Oscar.jl.

Usage

GroebnerWalk.jl provides its entire functionality through the function groebner_walk. The following example demonstrates the usage. First, we define the ideal using Oscar.

using Oscar

R, (x,y) = QQ[:x, :y]               # define ring ...
I = ideal([y^4+ x^3-x^2+x,x^4])     # ... and ideal

By default, groebner_walk starts with a Gröbner basis with respect to the default ordering on R and converts this into a Gröbner basis with respect to the lexicographic ordering on R. This is what the following code block accomplishes.

using GroebnerWalk

groebner_walk(I)                    # compute the Groebner basis

If one wants to specify target and start orderings explicitly, above function call needs to be written as follows.

groebner_walk(I, lex(R), default_ordering(R)) # compute the Groebner basis

For more detailed usage instructions and examples, please refer to the documentation.

Status

At the moment, the standard walk by Collart, Kalkbrener and Mall (1997) and the generic walk by Fukuda et al. (2007) are implemented.

Documentation

The documentation for the GroebnerWalk.jl package can be found at https://ooinaruhugh.github.io/GroebnerWalk.jl

Contacts

The library is maintained by Kamillo Ferry (kafe (at) kafe (dot) dev) and Francesco Nowell (francesconowell (at) gmail (dot) com).

Acknowledgement

The initial implementation was based on an implementation by Jordi Welp. We thank him for laying the groundwork for this package.

References

• Collart, S., M. Kalkbrener, and D. Mall. ‘Converting Bases with the Gröbner Walk’. Journal of Symbolic Computation 24, no. 3–4 (1997): 465–69.
• Cox, David A., John Little, and Donal O’Shea. Using Algebraic Geometry. Vol. 185. Graduate Texts in Mathematics. New York: Springer-Verlag, 2005.
• Fukuda, K., A. N. Jensen, N. Lauritzen, and R. Thomas. ‘The Generic Gröbner Walk’. Journal of Symbolic Computation 42, no. 3 (2007): 298–312.