fgb_sage — fgb_sage 0.2.2 documentation (2024)

Examples¶

See the examples and documentation athttps://fgb-sage.readthedocs.io.

Installation¶

Requirements: Linux with a recent version of Sage(tested with Sage 9.5.rc0 on Ubuntu 20.04 as well as with Sage 9.6 from Arch Linux).

First, clone the repository from GitHub and then compile andrun the tests:

git clone https://github.com/mwageringel/fgb_sage.git && cd fgb_sagesage setup.py test

After the tests passed successfully, run the following command to install thepackage for use with Sage:

sage -python -m pip install --upgrade --no-index -v .

Alternatively, to install into the Python user install directory (no rootaccess required), run:

sage -python -m pip install --upgrade --no-index -v --user .

Installing into a virtual environment, assuming Sage is installed system-wide:

python -m venv --system-site-packages ./sage-venv # assumes that sage and python-setuptools are available system-widesource ./sage-venv/bin/activate # redefines `python` and `pip` using virtual environmentpython -m pip install --upgrade --no-index -v .cd sage-venv && python -m IPython # changing directory is important In [1]: from sage.all import * In [2]: R = PolynomialRing(QQ, 5, 'x', order="degrevlex(2),degrevlex(3)") In [3]: I = sage.rings.ideal.Cyclic(R) In [4]: import fgb_sage In [5]: gb = fgb_sage.groebner_basis(I)

Documentation for fgb_sage¶

This interface has two main functions:

• groebner_basis() - compute a Groebner basis

• eliminate() - compute an elimination ideal

fgb_sage.groebner_basis(polys, **kwds)¶

Compute a Groebner basis of an ideal using FGb.

Supported term orders of the underlying polynomial ring are degrevlexorders, as well as block orders with two degrevlex blocks (eliminationorders). Supported coefficient fields are QQ and finite prime fields ofsize up to MAX_PRIME = 65521 < 2^16.

INPUT:

• polys – an ideal or a polynomial sequence, the generators of anideal.

• threads – integer (default: 1); only seems to work in positivecharacteristic.

• force_elim – integer (default: 0); if force_elim=1, then thecomputation will return only the result of the elimination, if anelimination order is used.

• verbosity – integer (default: 1), display progress info.

• matrix_bound – integer (default: 500000); this is is the maximalsize of the matrices generated by F4. This value can be increasedaccording to available memory.

• max_base – integer (default: 100000); maximum number ofpolynomials in output.

OUTPUT: the Groebner basis.

EXAMPLES:

This example computes a Groebner basis with respect to an eliminationorder:

sage: R = PolynomialRing(QQ, 5, 'x', order="degrevlex(2),degrevlex(3)")sage: I = sage.rings.ideal.Cyclic(R)sage: import fgb_sage # optional fgb_sagesage: gb = fgb_sage.groebner_basis(I) # optional fgb_sage, random...sage: gb.is_groebner(), gb.ideal() == I # optional fgb_sage(True, True)

Over finite fields, parallel computations are supported:

sage: R = PolynomialRing(GF(fgb_sage.MAX_PRIME), 4, 'x') # optional fgb_sagesage: I = sage.rings.ideal.Katsura(R) # optional fgb_sagesage: gb = fgb_sage.groebner_basis(I, threads=2, verbosity=0) # optional fgb_sage, randomsage: gb.is_groebner(), gb.ideal() == I # optional fgb_sage(True, True)

If fgb_sage.groebner_basis() is called with an ideal, the result iscached on MPolynomialIdeal.groebner_basis() so that othercomputations on the ideal do not need to recompute a Groebner basis:

sage: I.groebner_basis.is_in_cache() # optional fgb_sageTruesage: I.groebner_basis() is gb # optional fgb_sageTrue

However, note that gb.ideal() returns a new ideal and, thus, does nothave a Groebner basis in cache:

sage: gb.ideal().groebner_basis.is_in_cache() # optional fgb_sageFalse

TESTS:

Check that the result is not cached if it is only a basis of anelimination ideal:

sage: R = PolynomialRing(QQ, 5, 'x', order="degrevlex(2),degrevlex(3)")sage: I = sage.rings.ideal.Cyclic(R)sage: import fgb_sagesage: gb = fgb_sage.groebner_basis(I, force_elim=1) # optional fgb_sage, random...sage: I.groebner_basis.is_in_cache() # optional fgb_sageFalse

fgb_sage.eliminate(polys, elim_variables, **kwds)¶

Compute a Groebner basis with respect to an elimination order defined bythe given variables.

INPUT:

• polys – an ideal or a polynomial sequence.

• elim_variables – the variables to eliminate.

• force_elim – integer (default: 1).

• kwds – same as in groebner_basis().

OUTPUT: a Groebner basis of the elimination ideal.

EXAMPLES:

sage: R.<x,y,t,s,z> = PolynomialRing(QQ,5)sage: I = R * [x-t,y-t^2,z-t^3,s-x+y^3]sage: import fgb_sage # optional - fgb_sagesage: gb = fgb_sage.eliminate(I, [t,s], verbosity=0) # optional - fgb_sage, randomopen simulationsage: gb # optional - fgb_sage[x^2 - y, x*y - z, y^2 - x*z]sage: gb.is_groebner() # optional - fgb_sageTruesage: gb.ideal() == I.elimination_ideal([t,s]) # optional - fgb_sageTrue

Note

In some cases, this function fails to set the correct eliminationorder, see trac ticket #24981. This was fixed in Sage 8.7.

fgb_sage.internal_version()¶

Get the internal version of FGb.

OUTPUT: a dictionary containing the versions of FGb for FGb_int andFGb_modp, characteristic 0 and positive characteristic, respectively.These versions differ between Linux and macOS.

EXAMPLES:

sage: import fgb_sage # optional - fgb_sagesage: fgb_sage.internal_version() # optional - fgb_sage{'FGb_int': ..., 'FGb_modp': ...}

License¶

MIT for this package fgb_sage. However, note that FGb is licensed foracademic use only.

Links¶

• FGb homepage

• fgb_sage on GitHub

• documentation for Gröbner bases in Sage