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gfanGroebnerCone -- polyhedral information about a Groebner cone



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 :

For the programmer

The object gfanGroebnerCone is a method function with options.