Sage provides much of the functionality of gfan, which is a software package whose main function is to enumerate all reduced Groebner bases of a polynomial ideal. The reduced Groebner bases yield the maximal cones in the Groebner fan of the ideal. Several subcomputations can be issued and additional tools are included. Among these the highlights are:
AUTHORS:
EXAMPLES:
sage: x,y = QQ['x,y'].gens()
sage: i = ideal(x^2 - y^2 + 1)
sage: g = i.groebner_fan()
sage: g.reduced_groebner_bases()
[[x^2 - y^2 + 1], [-x^2 + y^2 - 1]]
TESTS:
sage: x,y = QQ['x,y'].gens()
sage: i = ideal(x^2 - y^2 + 1)
sage: g = i.groebner_fan()
sage: g == loads(dumps(g))
True
REFERENCES:
A utility function that takes a list of 4d polytopes, projects them to 3d, and returns a list of edges.
INPUT:
OUTPUT:
EXAMPLES:
sage: R4.<w,x,y,z> = PolynomialRing(QQ,4)
sage: gf = R4.ideal([w^2-x,x^2-y,y^2-z,z^2-1]).groebner_fan()
sage: g_cone = gf[0].groebner_cone()
sage: g_cone_facets = g_cone.facets()
sage: g_cone_ieqs = gf._cone_to_ieq(g_cone_facets)
sage: cone_data = ieq_to_vert(g_cone_ieqs,linearities=[[1,-1,-1,-1,-1]])
sage: cone_lines = gf._4d_to_3d(cone_data)
sage: cone_lines
[[[-3/5, -1/3, -1/5], [-1/7, 3/7, 5/7]], [[-3/5, -1/3, -1/5], [1, -1/3,
1/3]], [[-3/5, -1/3, -1/5], [1, 1, -1]], [[-1/7, 3/7, 5/7], [1, -1/3,
1/3]], [[-1/7, 3/7, 5/7], [1, 1, -1]], [[1, -1/3, 1/3], [1, 1, -1]]]
Tests equality of Groebner fan objects.
EXAMPLES:
sage: R.<q,u> = PolynomialRing(QQ,2)
sage: gf = R.ideal([q^2-u,u^2-q]).groebner_fan()
sage: gf2 = R.ideal([u^2-q,q^2-u]).groebner_fan()
sage: gf.__eq__(gf2)
True
Gets a reduced groebner basis
EXAMPLES;
sage: R4.<w1,w2,w3,w4> = PolynomialRing(QQ,4)
sage: gf = R4.ideal([w1^2-w2,w2^3-1,2*w3-w4^2,w4^2-w1]).groebner_fan()
sage: gf[0]
[w4^12 - 1, -1/2*w4^2 + w3, -w4^4 + w2, -w4^2 + w1]
INPUT:
EXAMPLES:
sage: R.<x,y,z> = PolynomialRing(QQ, 3, order='lex')
sage: I = R.ideal([x^2*y - z, y^2*z - x, z^2*x - y])
sage: G = I.groebner_fan(); G
Groebner fan of the ideal:
Ideal (x^2*y - z, -x + y^2*z, x*z^2 - y) of Multivariate Polynomial Ring in x, y, z over Rational Field
Returns an iterator for the reduced Groebner bases.
EXAMPLES:
sage: R.<x,y> = PolynomialRing(QQ,2)
sage: gf = R.ideal([x^3-y,y^3-x-1]).groebner_fan()
sage: a = gf.__iter__()
sage: a.next()
[y^9 - 3*y^6 + 3*y^3 - y - 1, -y^3 + x + 1]
A simple utility function for converting a facet normal to an inequality form.
EXAMPLES:
sage: R.<x,y> = PolynomialRing(QQ,2) # dummy stuff to get a gfan object
sage: gf = R.ideal([x^2+y,y^2]).groebner_fan()
sage: gf._cone_to_ieq([[1,2,3,4]])
[[0, 1, 2, 3, 4]]
Takes a 4-d vector and projects it onto the plane perpendicular to (1,1,1,1). Stretches by a factor of 2 as well, since this is only for graphical display purposes.
INPUT:
EXAMPLES:
sage: R.<x> = PolynomialRing(QQ,1) # dummy stuff to get a gfan object
sage: gf = R.ideal([x^2]).groebner_fan()
sage: gf._embed_tetra([1/2,1/2,1/2,1/2])
[0, 0, 0]
Return the ideal in gfan’s notation.
EXAMPLES:
sage: R.<x,y,z> = PolynomialRing(QQ,3)
sage: G = R.ideal([x^2*y - z, y^2*z - x, z^2*x - y]).groebner_fan()
sage: G._gfan_ideal()
'Q[x, y, z]{x^2*y-z,y^2*z-x,x*z^2-y}'
INPUT: none
OUTPUT:
EXAMPLES:
sage: R.<x,y,z> = PolynomialRing(QQ,3)
sage: G = R.ideal([x^2*y - z, y^2*z - x, z^2*x - y]).groebner_fan()
sage: G._gfan_maps()
(Ring morphism:
From: Multivariate Polynomial Ring in x, y, z over Rational Field
To: Multivariate Polynomial Ring in a, b, c over Rational Field
Defn: x |--> a
y |--> b
z |--> c,
Ring morphism:
From: Multivariate Polynomial Ring in a, b, c over Rational Field
To: Multivariate Polynomial Ring in x, y, z over Rational Field
Defn: a |--> x
b |--> y
c |--> z)
Return the extra options to the gfan command that are used by this object to account for working modulo a prime or in the presence of extra symmetries.
EXAMPLES:
sage: R.<x,y> = PolynomialRing(QQ,2)
sage: gf = R.ideal([x^3-y,y^3-x-1]).groebner_fan()
sage: gf._gfan_mod()
''
A string of the reduced Groebner bases of the ideal as output by gfan.
EXAMPLES:
sage: R.<a,b> = PolynomialRing(QQ,2)
sage: gf = R.ideal([a^3-b^2,b^2-a-1]).groebner_fan()
sage: gf._gfan_reduced_groebner_bases()
'Q[a,b]{{b^6-1+2*b^2-3*b^4,a+1-b^2},{b^2-1-a,a^3-1-a}}'
Return the ring in gfan’s notation
EXAMPLES:
sage: R.<x,y,z> = PolynomialRing(QQ,3)
sage: G = R.ideal([x^2*y - z, y^2*z - x, z^2*x - y]).groebner_fan()
sage: G._gfan_ring()
'Q[x, y, z]'
Return various statistics about this Groebner fan.
EXAMPLES:
sage: R.<x,y> = PolynomialRing(QQ)
sage: G = R.ideal([y^3 - x^2, y^2 - 13*x]).groebner_fan()
sage: G._gfan_stats()
{'Number of reduced Groebner bases': 3,
'Maximal total degree of a Groebner basis': 4,
'Dimension of homogeneity space': 0,
'Number of variables': 2,
'Minimal total degree of a Groebner basis': 2}
Describes the Groebner fan and its corresponding ideal.
EXAMPLES:
sage: R.<q,u> = PolynomialRing(QQ,2)
sage: gf = R.ideal([q-u,u^2-1]).groebner_fan()
sage: gf # indirect doctest
Groebner fan of the ideal:
Ideal (q - u, u^2 - 1) of Multivariate Polynomial Ring in q, u over Rational Field
Computes and returns a lexicographic reduced Groebner basis for the ideal.
EXAMPLES:
sage: R.<x,y,z> = PolynomialRing(QQ,3)
sage: G = R.ideal([x - z^3, y^2 - x + x^2 - z^3*x]).groebner_fan()
sage: G.buchberger()
[-z^3 + y^2, -z^3 + x]
Return the characteristic of the base ring.
EXAMPLES:
sage: R.<x,y,z> = PolynomialRing(QQ,3)
sage: i1 = ideal(x*z + 6*y*z - z^2, x*y + 6*x*z + y*z - z^2, y^2 + x*z + y*z)
sage: gf = i1.groebner_fan()
sage: gf.characteristic()
0
Return the dimension of the homogeneity space.
EXAMPLES:
sage: R.<x,y> = PolynomialRing(QQ,2)
sage: G = R.ideal([y^3 - x^2, y^2 - 13*x]).groebner_fan()
sage: G.dimension_of_homogeneity_space()
0
Returns the gfan output as a string given an input cmd; the default is to produce the list of reduced Groebner bases in gfan format.
EXAMPLES:
sage: R.<x,y> = PolynomialRing(QQ,2)
sage: gf = R.ideal([x^3-y,y^3-x-1]).groebner_fan()
sage: gf.gfan()
'Q[x,y]\n{{\ny^9-1-y+3*y^3-3*y^6,\nx+1-y^3}\n,\n{\ny^3-1-x,\nx^3-y}\n,\n{\ny-x^3,\nx^9-1-x}\n}\n'
Return the homogeneity space of a the list of polynomials that define this Groebner fan.
EXAMPLES:
sage: R.<x,y> = PolynomialRing(QQ,2)
sage: G = R.ideal([y^3 - x^2, y^2 - 13*x]).groebner_fan()
sage: H = G.homogeneity_space()
Return the ideal the was used to define this Groebner fan.
EXAMPLES:
sage: R.<x1,x2> = PolynomialRing(QQ,2)
sage: gf = R.ideal([x1^3-x2,x2^3-2*x1-2]).groebner_fan()
sage: gf.ideal()
Ideal (x1^3 - x2, x2^3 - 2*x1 - 2) of Multivariate Polynomial Ring in x1, x2 over Rational Field
See the documentation for self[0].interactive(). This does not work with the notebook.
EXAMPLES:
sage: print "This is not easily doc-testable; please write a good one!"
This is not easily doc-testable; please write a good one!
Return the maximal total degree of any 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.maximal_total_degree_of_a_groebner_basis()
4
Return the minimal total degree of any 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.minimal_total_degree_of_a_groebner_basis()
2
Return the number of reduced Groebner bases.
EXAMPLES:
sage: R.<x,y> = PolynomialRing(QQ,2)
sage: G = R.ideal([y^3 - x^2, y^2 - 13*x]).groebner_fan()
sage: G.number_of_reduced_groebner_bases()
3
Return the number of variables.
EXAMPLES:
sage: R.<x,y> = PolynomialRing(QQ,2)
sage: G = R.ideal([y^3 - x^2, y^2 - 13*x]).groebner_fan()
sage: G.number_of_variables()
2
sage: R = PolynomialRing(QQ,'x',10)
sage: R.inject_variables(globals())
Defining x0, x1, x2, x3, x4, x5, x6, x7, x8, x9
sage: G = ideal([x0 - x9, sum(R.gens())]).groebner_fan()
sage: G.number_of_variables()
10
Returns a polyhedral fan object corresponding to the reduced Groebner bases.
EXAMPLES:
sage: R3.<x,y,z> = PolynomialRing(QQ,3)
sage: gf = R3.ideal([x^8-y^4,y^4-z^2,z^2-1]).groebner_fan()
sage: pf = gf.polyhedralfan()
sage: pf.rays()
[[1, 0, 0], [0, 1, 0], [0, 0, 1]]
EXAMPLES:
sage: R.<x,y,z> = PolynomialRing(QQ, 3, order='lex')
sage: G = R.ideal([x^2*y - z, y^2*z - x, z^2*x - y]).groebner_fan()
sage: X = G.reduced_groebner_bases()
sage: len(X)
33
sage: X[0]
[z^15 - z, y - z^11, x - z^9]
sage: X[0].ideal()
Ideal (z^15 - z, y - z^11, x - z^9) of Multivariate Polynomial Ring in x, y, z over Rational Field
sage: X[:5]
[[z^15 - z, y - z^11, x - z^9],
[-y + z^11, y*z^4 - z, y^2 - z^8, x - z^9],
[-y^2 + z^8, y*z^4 - z, y^2*z^3 - y, y^3 - z^5, x - y^2*z],
[-y^3 + z^5, y*z^4 - z, y^2*z^3 - y, y^4 - z^2, x - y^2*z],
[-y^4 + z^2, y^6*z - y, y^9 - z, x - y^2*z]]
sage: R3.<x,y,z> = PolynomialRing(GF(2477),3)
sage: gf = R3.ideal([300*x^3-y,y^2-z,z^2-12]).groebner_fan()
sage: gf.reduced_groebner_bases()
[[z^2 - 12, y^2 - z, x^3 + 933*y],
[-y^2 + z, y^4 - 12, x^3 + 933*y],
[z^2 - 12, -300*x^3 + y, x^6 - 1062*z],
[-828*x^6 + z, -300*x^3 + y, x^12 + 200]]
Render a Groebner fan as sage graphics or save as an xfig file.
More precisely, the output is a drawing of the Groebner fan intersected with a triangle. The corners of the triangle are (1,0,0) to the right, (0,1,0) to the left and (0,0,1) at the top. If there are more than three variables in the ring we extend these coordinates with zeros.
INPUT:
EXAMPLES:
sage: R.<x,y,z> = PolynomialRing(QQ,3)
sage: G = R.ideal([y^3 - x^2, y^2 - 13*x,z]).groebner_fan()
sage: test_render = G.render()
sage: R.<x,y,z> = PolynomialRing(QQ,3)
sage: G = R.ideal([x^2*y - z, y^2*z - x, z^2*x - y]).groebner_fan()
sage: test_render = G.render(larger=True)
TESTS:
Testing the case where the number of generators is < 3. Currently, this should raise a NotImplementedError error.
sage: R.<x,y> = PolynomialRing(QQ, 2)
sage: R.ideal([y^3 - x^2, y^2 - 13*x]).groebner_fan().render()
...
NotImplementedError
For a Groebner fan of an ideal in a ring with four variables, this function intersects the fan with the standard simplex perpendicular to (1,1,1,1), creating a 3d polytope, which is then projected into 3 dimensions. The edges of this projected polytope are returned as lines.
EXAMPLES:
sage: R4.<w,x,y,z> = PolynomialRing(QQ,4)
sage: gf = R4.ideal([w^2-x,x^2-y,y^2-z,z^2-x]).groebner_fan()
sage: three_d = gf.render3d()
TESTS:
Now test the case where the number of generators is not 4. Currently, this should raise a NotImplementedError error.
sage: P.<a,b,c> = PolynomialRing(QQ, 3, order="lex")
sage: sage.rings.ideal.Katsura(P, 3).groebner_fan().render3d()
...
NotImplementedError
Return the multivariate polynomial ring.
EXAMPLES:
sage: R.<x1,x2> = PolynomialRing(QQ,2)
sage: gf = R.ideal([x1^3-x2,x2^3-x1-2]).groebner_fan()
sage: gf.ring()
Multivariate Polynomial Ring in x1, x2 over Rational Field
Return a tropical basis for the tropical curve associated to this ideal.
INPUT:
EXAMPLES:
sage: R.<x,y,z> = PolynomialRing(QQ,3, order='lex')
sage: G = R.ideal([y^3-3*x^2, z^3-x-y-2*y^3+2*x^2]).groebner_fan()
sage: G
Groebner fan of the ideal:
Ideal (-3*x^2 + y^3, 2*x^2 - x - 2*y^3 - y + z^3) of Multivariate Polynomial Ring in x, y, z over Rational Field
sage: G.tropical_basis()
[-3*x^2 + y^3, 2*x^2 - x - 2*y^3 - y + z^3, 3/4*x + y^3 + 3/4*y - 3/4*z^3]
Returns information about the tropical intersection of the polynomials defining the ideal.
EXAMPLES:
sage: R.<x,y,z> = PolynomialRing(QQ,3)
sage: i1 = ideal(x*z + 6*y*z - z^2, x*y + 6*x*z + y*z - z^2, y^2 + x*z + y*z)
sage: gf = i1.groebner_fan()
sage: pf = gf.tropical_intersection()
sage: pf.rays()
[[-1, 0, 0]]
Returns the weight vectors corresponding to the reduced Groebner bases.
EXAMPLES:
sage: r3.<x,y,z> = PolynomialRing(QQ,3)
sage: g = r3.ideal([x^3+y,y^3-z,z^2-x]).groebner_fan()
sage: g.weight_vectors()
[(3, 7, 1), (5, 1, 2), (7, 1, 4), (1, 1, 4), (1, 1, 1), (1, 4, 1), (1, 4, 10)]
sage: r4.<x,y,z,w> = PolynomialRing(QQ,4)
sage: g4 = r4.ideal([x^3+y,y^3-z,z^2-x,z^3 - w]).groebner_fan()
sage: len(g4.weight_vectors())
23
Converts polymake/gfan data on a polyhedral cone into a sage class. Currently (18-03-2008) needs a lot of work.
EXAMPLES:
sage: R3.<x,y,z> = PolynomialRing(QQ,3)
sage: gf = R3.ideal([x^8-y^4,y^4-z^2,z^2-2]).groebner_fan()
sage: a = gf[0].groebner_cone()
sage: a.facets()
[[0, 0, 1], [0, 1, 0], [1, 0, 0]]
Returns a basic description of the polyhedral cone.
EXAMPLES:
sage: R3.<x,y,z> = PolynomialRing(QQ,3)
sage: gf = R3.ideal([x^8-y^4,y^4-z^2,z^2-2]).groebner_fan()
sage: a = gf[0].groebner_cone()
sage: a # indirect doctests
Polyhedral cone in 3 dimensions of dimension 3
Returns the ambient dimension of the Groebner cone.
EXAMPLES:
sage: R3.<x,y,z> = PolynomialRing(QQ,3)
sage: gf = R3.ideal([x^8-y^4,y^4-z^2,z^2-2]).groebner_fan()
sage: a = gf[0].groebner_cone()
sage: a.ambient_dim()
3
Returns the dimension of the Groebner cone.
EXAMPLES:
sage: R3.<x,y,z> = PolynomialRing(QQ,3)
sage: gf = R3.ideal([x^8-y^4,y^4-z^2,z^2-2]).groebner_fan()
sage: a = gf[0].groebner_cone()
sage: a.dim()
3
Returns the inward facet normals of the Groebner cone.
EXAMPLES:
sage: R3.<x,y,z> = PolynomialRing(QQ,3)
sage: gf = R3.ideal([x^8-y^4,y^4-z^2,z^2-2]).groebner_fan()
sage: a = gf[0].groebner_cone()
sage: a.facets()
[[0, 0, 1], [0, 1, 0], [1, 0, 0]]
Returns the lineality dimension of the Groebner cone. This is just the difference between the ambient dimension and the dimension of the cone.
EXAMPLES:
sage: R3.<x,y,z> = PolynomialRing(QQ,3)
sage: gf = R3.ideal([x^8-y^4,y^4-z^2,z^2-2]).groebner_fan()
sage: a = gf[0].groebner_cone()
sage: a.lineality_dim()
0
Returns a point in the relative interior of the Groebner cone.
EXAMPLES:
sage: R3.<x,y,z> = PolynomialRing(QQ,3)
sage: gf = R3.ideal([x^8-y^4,y^4-z^2,z^2-2]).groebner_fan()
sage: a = gf[0].groebner_cone()
sage: a.relative_interior_point()
[1, 1, 1]
Converts polymake/gfan data on a polyhedral fan into a sage class. Currently (18-03-2008) needs a lot of work.
INPUT:
EXAMPLES:
sage: R.<x,y,z> = PolynomialRing(QQ,3)
sage: i2 = ideal(x*z + 6*y*z - z^2, x*y + 6*x*z + y*z - z^2, y^2 + x*z + y*z)
sage: gf2 = i2.groebner_fan(verbose = False)
sage: pf = gf2.polyhedralfan()
sage: pf.rays()
[[1, 0, 0], [-2, -1, 0], [1, 1, 0], [0, -1, 0], [-1, 1, 0]]
Returns a basic description of the polyhedral fan.
EXAMPLES:
sage: R3.<x,y,z> = PolynomialRing(QQ,3)
sage: gf = R3.ideal([x^8-y^4,y^4-z^2,z^2-2]).groebner_fan()
sage: pf = gf.polyhedralfan()
sage: pf # indirect doctest
Polyhedral fan in 3 dimensions of dimension 3
Returns the raw output of gfan as a string. This should only be needed internally as all relevant output is converted to sage objects.
EXAMPLES:
sage: R3.<x,y,z> = PolynomialRing(QQ,3)
sage: gf = R3.ideal([x^8-y^4,y^4-z^2,z^2-2]).groebner_fan()
sage: pf = gf.polyhedralfan()
sage: pf._str_()
'_application PolyhedralFan\n_version 2.2\n_type PolyhedralFan\n\nAMBIENT_DIM\n3\n\nDIM\n3\n\nLINEALITY_DIM\n0\n\nRAYS\n1 0 0\t# 0\n0 1 0\t# 1\n0 0 1\t# 2\n\nN_RAYS\n3\n\nLINEALITY_SPACE\n\nORTH_LINEALITY_SPACE\n0 0 1\n0 1 0\n1 0 0\n\nF_VECTOR\n1 3 3 1\n\nCONES\n{}\t# Dimension 0\n{0}\t# Dimension 1\n{1}\n{2}\n{0 1}\t# Dimension 2\n{0 2}\n{1 2}\n{0 1 2}\t# Dimension 3\n\nMAXIMAL_CONES\n{0 1 2}\t# Dimension 3\n\nPURE\n1\n'
Returns the ambient dimension of the Groebner fan.
EXAMPLES:
sage: R3.<x,y,z> = PolynomialRing(QQ,3)
sage: gf = R3.ideal([x^8-y^4,y^4-z^2,z^2-2]).groebner_fan()
sage: a = gf.polyhedralfan()
sage: a.ambient_dim()
3
Returns the dimension of the Groebner fan.
EXAMPLES:
sage: R3.<x,y,z> = PolynomialRing(QQ,3)
sage: gf = R3.ideal([x^8-y^4,y^4-z^2,z^2-2]).groebner_fan()
sage: a = gf.polyhedralfan()
sage: a.dim()
3
Returns the lineality dimension of the Groebner fan. This is just the difference between the ambient dimension and the dimension of the cone.
EXAMPLES:
sage: R3.<x,y,z> = PolynomialRing(QQ,3)
sage: gf = R3.ideal([x^8-y^4,y^4-z^2,z^2-2]).groebner_fan()
sage: a = gf.polyhedralfan()
sage: a.lineality_dim()
0
Returns a list of rays of the polyhedral fan.
EXAMPLES:
sage: R.<x,y,z> = PolynomialRing(QQ,3)
sage: i2 = ideal(x*z + 6*y*z - z^2, x*y + 6*x*z + y*z - z^2, y^2 + x*z + y*z)
sage: gf2 = i2.groebner_fan(verbose = False)
sage: pf = gf2.polyhedralfan()
sage: pf.rays()
[[1, 0, 0], [-2, -1, 0], [1, 1, 0], [0, -1, 0], [-1, 1, 0]]
A class for representing reduced Groebner bases as produced by gfan.
INPUT:
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 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
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}'
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]
Return defining inequalities for the full-dimensional Groebner cone associated to this marked minimal reduced Groebner basis.
INPUT:
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]]
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
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 *
*********************************************
....
Utility function that parses cone information into a dict indexed by dimension.
INPUT:
EXAMPLES:
sage: R.<x,y,z,w> = PolynomialRing(QQ,4)
sage: ts = R.ideal([x^2+y^2+z^2-1,x^4-y^2-z-1,x+y+z,w+x+y])
sage: tsg = ts.groebner_fan()
sage: tstr = tsg.tropical_intersection()
sage: from sage.rings.polynomial.groebner_fan import _cone_parse
sage: _cone_parse(tstr.fan_dict['CONES'])
{1: [[0], [1], [3], [2], [4]], 2: [[2, 4]]}
Computes the maximum degree of a list of polynomials
EXAMPLES:
sage: from sage.rings.polynomial.groebner_fan import max_degree
sage: R.<x,y> = PolynomialRing(QQ,2)
sage: p_list = [x^2-y,x*y^10-x]
sage: max_degree(p_list)
11.0
Checks if any strings in a list are prefixes of another string in the list.
EXAMPLES:
sage: from sage.rings.polynomial.groebner_fan import prefix_check
sage: prefix_check(['z1','z1z1'])
False
sage: prefix_check(['z1','zz1'])
True