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Package sage :: Package rings :: Package polynomial :: Module groebner_fan :: Class ReducedGroebnerBasis
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Class ReducedGroebnerBasis

source code

                      object --+    
                               |    
structure.sage_object.SageObject --+
                                   |
                      object --+   |
                               |   |
                            list --+
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                                  ReducedGroebnerBasis
Instance Methods [hide private]
 
__init__(self, groebner_fan, gens, gfan_gens)
A class for representing reduced Groebner bases as produced by gfan.		 source code
 
_repr_(self)
Returns the reduced Groebner basis as a string.		 source code
 
_gfan_gens(self)
Returns the reduced Groebner basis as a string in gfan format.		 source code
 
_gfan(self)
Returns a description of the Groebner fan this basis was derived from.		 source code
 
interactive(self, latex=False, flippable=False, wall=False, inequalities=False, weight=False)
Do an interactive walk of the Groebner fan starting at this reduced Groebner basis.		 source code
 
groebner_cone(self, restrict=False)
Return defining inequalities for the full-dimensional Groebner cone associated to this marked minimal reduced Groebner basis.		 source code
 
ideal(self)
Return the ideal generated by this basis.		 source code

Inherited from structure.sage_object.SageObject: __hash__, __new__, __repr__, _axiom_, _axiom_init_, _gap_, _gap_init_, _gp_, _gp_init_, _interface_, _interface_init_, _interface_is_cached_, _kash_, _kash_init_, _macaulay2_, _macaulay2_init_, _magma_, _magma_init_, _maple_, _maple_init_, _mathematica_, _mathematica_init_, _maxima_, _maxima_init_, _octave_, _octave_init_, _pari_, _pari_init_, _r_init_, _sage_, _singular_, _singular_init_, category, db, dump, dumps, plot, rename, reset_name, save, version

Inherited from list: __add__, __contains__, __delitem__, __delslice__, __eq__, __ge__, __getattribute__, __getitem__, __getslice__, __gt__, __iadd__, __imul__, __iter__, __le__, __len__, __lt__, __mul__, __ne__, __reversed__, __rmul__, __setitem__, __setslice__, append, count, extend, index, insert, pop, remove, reverse, sort

Inherited from object: __delattr__, __reduce__, __reduce_ex__, __setattr__, __str__

Properties [hide private]

Inherited from object: __class__

Method Details [hide private]

__init__(self, groebner_fan, gens, gfan_gens)
(Constructor)

source code  

A class for representing reduced Groebner bases as produced by gfan.

INPUT:
    groebner_fan -- a GroebnerFan object from an ideal
    gens -- the generators of the ideal
    gfan_gens -- the generators as a gfan string
    
EXAMPLES:
    sage: R.<a,b> = PolynomialRing(QQ,2)
    sage: gf = R.ideal([a^2-b^2,b-a-1]).groebner_fan()
    sage: from sage.rings.polynomial.groebner_fan import ReducedGroebnerBasis
    sage: ReducedGroebnerBasis(gf,gf[0],gf[0]._gfan_gens())
    [b - 1/2, a + 1/2]
Returns:
new list
Overrides: list.__init__

_repr_(self)

source code  

Returns the reduced Groebner basis as a string.

EXAMPLES:
    sage: R.<z1,zz1> = PolynomialRing(QQ,2)
    sage: gf = R.ideal([z1^2*zz1-1,zz1-2]).groebner_fan()
    sage: rgb1 = gf.reduced_groebner_bases()[0]
    sage: rgb1 # indirect doctest
    [zz1 - 2, z1^2 - 1/2]

_gfan_gens(self)

source code  

Returns the reduced Groebner basis as a string in gfan format.

EXAMPLES:
    sage: R.<z1,zz1> = PolynomialRing(QQ,2)
    sage: gf = R.ideal([z1^2*zz1-1,zz1-2]).groebner_fan()
    sage: rgb1 = gf.reduced_groebner_bases()[0]
    sage: rgb1._gfan_gens()
    '{zz1-2,z1^2-1/2}'

_gfan(self)

source code  

Returns a description of the Groebner fan this basis was derived from.

EXAMPLES:
    sage: R.<z1,zz1> = PolynomialRing(QQ,2)
    sage: gf = R.ideal([z1^2*zz1-1,zz1-2]).groebner_fan()
    sage: rgb1 = gf.reduced_groebner_bases()[0]
    sage: rgb1._gfan()
    Groebner fan of the ideal:
    Ideal (z1^2*zz1 - 1, zz1 - 2) of Multivariate Polynomial Ring in z1, zz1 over Rational Field

interactive(self, latex=False, flippable=False, wall=False, inequalities=False, weight=False)

source code  

Do an interactive walk of the Groebner fan starting at this
reduced Groebner basis.

EXAMPLES:
    sage: R.<x,y> = PolynomialRing(QQ,2)
    sage: G = R.ideal([y^3 - x^2, y^2 - 13*x]).groebner_fan()
    sage: G[0].interactive()      # not tested
    Initializing gfan interactive mode
    *********************************************
    *     Press control-C to return to SAGE     *
    *********************************************
    ....

groebner_cone(self, restrict=False)

source code  

Return defining inequalities for the full-dimensional
Groebner cone associated to this marked minimal reduced
Groebner basis.

INPUT:
    restrict -- bool (default: False); if True, add an inequality for
                each coordinate, so that the cone is restricted to
                the positive orthant.

OUTPUT:
    tuple of integer vectors

EXAMPLES:
    sage: R.<x,y> = PolynomialRing(QQ,2)
    sage: G = R.ideal([y^3 - x^2, y^2 - 13*x]).groebner_fan()
    sage: poly_cone = G[1].groebner_cone()
    sage: poly_cone.facets()
    [[-1, 2], [1, -1]]
    sage: [g.groebner_cone().facets() for g in G]
    [[[0, 1], [1, -2]], [[-1, 2], [1, -1]], [[-1, 1], [1, 0]]]
    sage: G[1].groebner_cone(restrict=True).facets()
    [[-1, 2], [1, -1]]

ideal(self)

source code  

Return the ideal generated by this basis.

EXAMPLES:
    sage: R.<x,y,z> = PolynomialRing(QQ,3)
    sage: G = R.ideal([x - z^3, y^2 - 13*x]).groebner_fan()
    sage: G[0].ideal()
    Ideal (-13*z^3 + y^2, -z^3 + x) of Multivariate Polynomial Ring in x, y, z over Rational Field

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