S = gfanGroebnerCone(M)
S = gfanGroebnerCone(L)
S = gfanGroebnerCone(I)
S = gfanGroebnerCone(K, M)
S = gfanGroebnerCone(K, L)
S = gfanGroebnerCone(K, I)
S = gfanGroebnerCone(N, M)
S = gfanGroebnerCone(N, L)
S = gfanGroebnerCone(N, I)
S = gfanGroebnerCone(J, M)
S = gfanGroebnerCone(J, L)
S = gfanGroebnerCone(J, I)
This method computes the Grobener cone of the input in the case where M, L, I are reduced Groebner bases. If M, L, I are only minimal bases, then a smaller cone is produced.


In the above example any weights w = a(1,1) + p (1,1) for a a real number and p >= 0 give (x) as the initial ideal of (x+y) with respect to w.
When both K and M are given as input and are compatible marked reduced Groebner bases in the sense that K is an initial ideal of M then gfanGroebnerCone(K,M) computes the cone of K in the fan of M. For example, the cone on which (x+y) is its own initial ideal is simply the line w = a(1,1) for a a real number.


Note that the pair option will automatically be specified when two marked Groebner bases are given.
gfan Documentation
This program computes a Groebner cone. Three different cases are handled. The input may be a marked reduced Groebner basis in which case its Groebner cone is computed. The input may be just a marked minimal basis in which case the cone computed is not a Groebner cone in the usual sense but smaller. (These cones are described in [Fukuda, Jensen, Lauritzen, Thomas]). The third possible case is that the Groebner cone is possibly lower dimensional and given by a pair of Groebner bases as it is useful to do for tropical varieties, see option pair. The facets of the cone can be read off in section FACETS and the equations in section IMPLIED_EQUATIONS.
Options:
restrict:
Add an inequality for each coordinate, so that the the cone is restricted to the nonnegative orthant.
pair:
The Groebner cone is given by a pair of compatible Groebner bases. The first basis is for the initial ideal and the second for the ideal itself. See the tropical section of the manual.
asfan:
Writes the cone as a polyhedral fan with all its faces instead. In this way the extreme rays of the cone are also computed.
vectorinput:
Compute a cone given list of inequalities rather than a Groebner cone. The input is an integer which specifies the dimension of the ambient space, a list of inequalities given as vectors and a list of equations.
The object gfanGroebnerCone is a method function with options.