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 # optional - giacpy_sage
sage: P = PolynomialRing(QQ, 6, 'x')
sage: I = sage.rings.ideal.Cyclic(P)
sage: B = gb_giac(I.gens()) # optional - giacpy_sage, random
sage: B # optional - giacpy_sage
Polynomial Sequence with 45 Polynomials in 6 Variables
class sage.libs.giac.GiacSettingsDefaultContext

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 # optional - giacpy_sage
sage: P = PolynomialRing(GF(previous_prime(2**31)), 6, 'x') # optional - giacpy_sage
sage: I = sage.rings.ideal.Cyclic(P) # optional - giacpy_sage
sage: B=gb_giac(I.gens());B # optional - giacpy_sage

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

Elimination ideals can be computed by passing elim_variables:

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

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

Computations over QQ can benefit from

  • a probabilistic lifting:

    sage: P = PolynomialRing(QQ,5, 'x') # optional - giacpy_sage
    sage: I = ideal([P.random_element(3,7) for j in range(5)]) # optional - giacpy_sage
    sage: B1 = gb_giac(I.gens(),1e-16) # optional - giacpy_sage, long time (1s)
    ...adding reconstructed ideal generators...
    ...
    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) # optional - giacpy_sage
    sage: B2 = gb_giac(I.gens()) # optional - giacpy_sage, long time (4s)
    
    // Groebner basis computation time...
    sage: B1 == B2 # optional - giacpy_sage, long time
    True
    sage: B1.is_groebner() # optional - giacpy_sage, long time (20s)
    True
    
  • multi threaded operations:

    sage: P = PolynomialRing(QQ, 8, 'x') # optional - giacpy_sage
    sage: I = sage.rings.ideal.Cyclic(P) # optional - giacpy_sage
    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) # optional - giacpy_sage
sage: gb_giac(I,prot=True)  # optional - giacpy_sage, 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):  # optional - giacpy_sage
....:    giacsettings.proba_epsilon = a/100
....:    giacsettings.threads = b+2
....:    return (giacsettings.proba_epsilon, giacsettings.threads)

sage: from giacpy_sage import giacsettings  # optional - giacpy_sage
sage: from sage.libs.giac import local_giacsettings  # optional - giacpy_sage
sage: gporig, gtorig = (giacsettings.proba_epsilon,giacsettings.threads)  # optional - giacpy_sage
sage: gp, gt = local_giacsettings(testf)(giacsettings.proba_epsilon,giacsettings.threads)  # optional - giacpy_sage
sage: gporig == giacsettings.proba_epsilon  # optional - giacpy_sage
True
sage: gtorig == giacsettings.threads  # optional - giacpy_sage
True
sage: gp<gporig, gt-gtorig  # optional - giacpy_sage
(True, 2)