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gfanToPolyhedralFan -- polyhedral data about the fan of a list of Groebner bases



This method takes a list of marked reduced Groebner bases and outputs a polyhedral fan whose faces correspond to the marked Groebner bases.

For example, $x + y$ is a marked Groebner basis with initial term $x$ for any weight vector $(a,b)$ with $a < b$.

i1 : QQ[x,y]

o1 = QQ[x..y]

o1 : PolynomialRing
i2 : M = markedPolynomialList {{x},{x+y}}

o2 = {(x) + y}

o2 : MarkedPolynomialList
i3 : F = gfanToPolyhedralFan { M }

o3 = F

o3 : Fan

The single facet above is $\mathbb R_{\geq 0} (1,-1) + \mathbb R (1,1)$ from reading off the ray and the lineality space. This corresponds to $\{(a,b) \mid a < b\}$.

Adding in the Groebner basis with $y$ as the initial term gives a second facet.

i4 : L = markedPolynomialList {{y},{x+y}}

o4 = {(y) + x}

o4 : MarkedPolynomialList
i5 : F = gfanToPolyhedralFan { M, L }

o5 = F

o5 : Fan

gfan Documentation This program takes a list of reduced Groebner bases and produces the fan of all faces of these. In this way by giving the complete list of reduced Groebner bases, the Groebner fan can be computed as a polyhedral complex. The option --restrict lets the user choose between computing the Groebner fan or the restricted Groebner fan.Options:--restrict: Add an inequality for each coordinate, so that the the cones are restricted to the non-negative orthant.--symmetry: Tell the program to read in generators for a group of symmetries (subgroup of $S_n$) after having read in the ring. The output is grouped according to these symmetries. Only one representative for each orbit is needed on the input.

Ways to use gfanToPolyhedralFan :

For the programmer

The object gfanToPolyhedralFan is a method function with options.