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groebnerFan -- the fan of all groebner bases of an ideal

Synopsis

Usage:
P = groebnerFan(I)
Inputs:
- I, an ideal, of which to compute the fan
- Symmetries, a list, of permutations of the variables leaving the ideal invariant
Outputs:
- P, a hash table, containing all the data of the polyhedral fan of I.
Optional inputs:
- Symmetries => ..., -- the fan of all groebner bases of an ideal

Description

i1 : R = QQ[symbol a..symbol f];

i2 : I = pfaffians(4, genericSkewMatrix(R,4))

o2 = ideal(c*d - b*e + a*f)

o2 : Ideal of R

i3 : P = groebnerFan(I)
using temporary file /tmp/M2-83082-1

o3 = PolymakeObject{...11...}

o3 : PolymakeObject

i4 : peek P

o4 = PolymakeObject{AMBIENT_DIM => 6                                            }
                    CONES => {{}, {0}, {1}, {2}, {0, 2}, {0, 1}, {1, 2}}
                    DIM => 6
                    F_VECTOR => {1, 3, 3}
                    LINEALITY_DIM => 4
                    LINEALITY_SPACE => | 0 0 0 1 1  1  |
                                       | 0 0 1 0 1  1  |
                                       | 0 1 0 0 -1 0  |
                                       | 1 0 0 0 0  -1 |
                    MAXIMAL_CONES => {{}, {0}, {1}, {2}, {0, 2}, {0, 1}, {1, 2}}
                    N_RAYS => 3
                    ORTH_LINEALITY_SPACE => | 0 1 -1 -1 1 0 |
                                            | 1 0 -1 -1 0 1 |
                    PURE => 1
                    RAYS => | -1 0  0 0 0 0 |
                            | 1  1  0 0 0 0 |
                            | 0  -1 0 0 0 0 |

Caveat

Requires loading of the Polymake package to make the PolymakeObject type available.

See also

gfan -- all initial ideals of an ideal
initialIdeal -- initial ideal with respect to a weight vector
groebnerCone -- the cone whose interior weight vectors give the given initial ideal
Symmetries -- permutations leaving an ideal invariant to speed up gfan computations.

Ways to use `groebnerFan` :

groebnerFan(Ideal)