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# gfanGroebnerCone -- polyhedral information about a Groebner cone

## Synopsis

• Usage:
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)
• Inputs:
• K, , a marked reduced Groebner basis, or a minimal basis.
• M, , a marked reduced Groebner basis.
• N, a list, a list of reduced Groebner basis, or a minimal basis.
• L, a list, a list of reduced Groebner basis.
• J, an ideal, an ideal of reduced Groebner basis, or a minimal basis.
• I, an ideal, an ideal of reduced Grobner basis.
• Optional inputs:
• asfan
• pair
• restrict
• vectorinput
• xml
• Outputs:
• S, , a description of the Groebner cone of M or I or L

## Description

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.

 i1 : QQ[x,y]; i2 : gfanGroebnerCone( markedPolynomialList {{x}, {x+y}} ) o2 = Fan{...1...} o2 : Fan

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.

 i3 : QQ[x,y]; i4 : gfanGroebnerCone( markedPolynomialList {{x}, {x+y}}, markedPolynomialList {{x}, {x+y}} ) o4 = Fan{...1...} o4 : Fan

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 non-negative 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.


## Ways to use gfanGroebnerCone :

• gfanGroebnerCone(Ideal)
• gfanGroebnerCone(Ideal,Ideal)
• gfanGroebnerCone(Ideal,List)
• gfanGroebnerCone(Ideal,MarkedPolynomialList)
• gfanGroebnerCone(List)
• gfanGroebnerCone(List,Ideal)
• gfanGroebnerCone(List,List)
• gfanGroebnerCone(List,MarkedPolynomialList)
• gfanGroebnerCone(MarkedPolynomialList)
• gfanGroebnerCone(MarkedPolynomialList,Ideal)
• gfanGroebnerCone(MarkedPolynomialList,List)
• gfanGroebnerCone(MarkedPolynomialList,MarkedPolynomialList)

## For the programmer

The object gfanGroebnerCone is .