# Wrappers for Giac functions¶

We provide a python function to compute and convert to sage a Groebner basis using the giacpy_sage module.

AUTHORS:

• Martin Albrecht (2015-07-01): initial version

• Han Frederic (2015-07-01): initial version

EXAMPLES:

sage: from sage.libs.giac import groebner_basis as gb_giac # random
sage: P = PolynomialRing(QQ, 6, 'x')
sage: I = sage.rings.ideal.Cyclic(P)
sage: B = gb_giac(I.gens()) # random
sage: B
Polynomial Sequence with 45 Polynomials in 6 Variables

class sage.libs.giac.GiacSettingsDefaultContext

Bases: object

Context preserve libgiac settings.

sage.libs.giac.groebner_basis(gens, proba_epsilon=None, threads=None, prot=False, elim_variables=None, *args, **kwds)

Compute a Groebner Basis of an ideal using giacpy_sage. The result is automatically converted to sage.

Supported term orders of the underlying polynomial ring are lex, deglex, degrevlex and block orders with 2 degrevlex blocks.

INPUT:

• gens - an ideal (or a list) of polynomials over a prime field of characteristic 0 or p<2^31

• proba_epsilon - (default: None) majoration of the probability

of a wrong answer when probabilistic algorithms are allowed.

• if proba_epsilon is None, the value of sage.structure.proof.all.polynomial() is taken. If it is false then the global giacpy_sage.giacsettings.proba_epsilon is used.

• if proba_epsilon is 0, probabilistic algorithms are disabled.

• threads - (default: None) Maximal number of threads allowed for giac. If None, the global giacpy_sage.giacsettings.threads is considered.

• prot - (default: False) if True print detailled informations

• elim_variables - (default: None) a list of variables to eliminate from the ideal.

• if elim_variables is None, a Groebner basis with respect to the term ordering of the parent polynomial ring of the polynomials gens is computed.

• if elim_variables is a list of variables, a Groebner basis of the elimination ideal with respect to a degrevlex term order is computed, regardless of the term order of the polynomial ring.

OUTPUT:

Polynomial sequence of the reduced Groebner basis.

EXAMPLES:

sage: from sage.libs.giac import groebner_basis as gb_giac
sage: P = PolynomialRing(GF(previous_prime(2**31)), 6, 'x')
sage: I = sage.rings.ideal.Cyclic(P)
sage: B=gb_giac(I.gens());B

// Groebner basis computation time ...
Polynomial Sequence with 45 Polynomials in 6 Variables
sage: B.is_groebner()
True


Elimination ideals can be computed by passing elim_variables:

sage: P = PolynomialRing(GF(previous_prime(2**31)), 5, 'x')
sage: I = sage.rings.ideal.Cyclic(P)
sage: B = gb_giac(I.gens(), elim_variables=[P.gen(0), P.gen(2)])

// Groebner basis computation time ...
sage: B.is_groebner()
True
sage: B.ideal() == I.elimination_ideal([P.gen(0), P.gen(2)])
True


Computations over QQ can benefit from

• a probabilistic lifting:

sage: P = PolynomialRing(QQ,5, 'x')
sage: I = ideal([P.random_element(3,7) for j in range(5)])
sage: B1 = gb_giac(I.gens(),1e-16) # long time (1s)
...Running a probabilistic check for the reconstructed Groebner basis.
If successfull, error probability is less than 1e-16 ...
sage: sage.structure.proof.all.polynomial(True)
sage: B2 = gb_giac(I.gens()) # long time (4s)

// Groebner basis computation time...
sage: B1 == B2 # long time
True
sage: B1.is_groebner() # long time (20s)
True


sage: P = PolynomialRing(QQ, 8, 'x')
sage: I = sage.rings.ideal.Cyclic(P)
sage: time B = gb_giac(I.gens(),1e-6,threads=2) # doctest: +SKIP
Running a probabilistic check for the reconstructed Groebner basis...
Time: CPU 168.98 s, Wall: 94.13 s


You can get detailled information by setting prot=True

sage: I = sage.rings.ideal.Katsura(P)
sage: gb_giac(I,prot=True)  # random, long time (3s)
9381383 begin computing basis modulo 535718473
9381501 begin new iteration zmod, number of pairs: 8, base size: 8
...end, basis size 74 prime number 1
G=Vector [0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,...
...creating reconstruction #0
...
++++++++basis size 74
checking pairs for i=0, j=
checking pairs for i=1, j=2,6,12,17,19,24,29,34,39,42,43,48,56,61,64,69,
...
checking pairs for i=72, j=73,
checking pairs for i=73, j=
Number of critical pairs to check 373
+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++...
Successfull check of 373 critical pairs
12380865 end final check
Polynomial Sequence with 74 Polynomials in 8 Variables

sage.libs.giac.local_giacsettings(func)

Decorator to preserve Giac’s proba_epsilon and threads settings.

EXAMPLES:

sage: def testf(a,b):
....:    giacsettings.proba_epsilon = a/100