colorfulSOP -- Colorful System of Parameters (KSOP)

Synopsis

Usage:

colorfulSOP(B,n)

colorfulSOP(D)
Inputs:
- B, an abstract simplicial complex, the barycentric subdivision of a simplicial complex D
- n, an integer, the number of vertices of D
- D, an abstract simplicial complex, any simplicial complex for which you want the KSOP for its barycentric subdivision
Outputs:
- a list, a list of sums of ring elements of the polynomial ring of B

Description

The barycentric subdivision of a simplicial complex is a balanced simplicial complex, and so in this version we require the input of the barycentric subdivision of a simplicial complex.

The examples use the simplicial complex D on five vertices consisting of an isolated vertex and triangle of dimension 2 attached to an edge. The first example uses the barycentric subdivision of D. When considering the barycentric subdivision of a simplicial complex it is often useful to remove the minimal element of its face poset. In the next example, we demonstrate how this can be accomplished.

i1 : S = QQ[a..e];

i2 : D = simplicialComplex {a*b*c,c*d,e};

i3 : n = # vertices D

o3 = 5

i4 : ComplexIncludingMinElt = barycentricSubdivision D;

i5 : R = ring(ComplexIncludingMinElt);

i6 : B = faceDelete(R_0,ComplexIncludingMinElt);

i7 : colorfulSOP(B,n)

o7 = {v  + v  + v  + v  + v , v  + v  + v  + v , v  }
       1    2    3    4    5   6    7    8    9   10

o7 : List

Also included is the option to input any simplicial complex D in order to obtain the colorful system of parameters for the face ring of the corresponding barycentric subdivision of D.

i8 : S = QQ[a..e];

i9 : D = simplicialComplex {a*b*c,c*d,e};

i10 : colorfulSOP(D)

o10 = {v  + v  + v  + v  + v , v  + v  + v  + v , v  }
        1    2    3    4    5   6    7    8    9   10

o10 : List

Caveat

This current version requires the input of the barycentric subdivision of a simplicial complex D without the minimal element of the face poset, which corresponds to the empty set of D. This must be accomplished by using the barycentricSubdivision method, included in this package. It also has the option of inputting any simplicial complex. The system then takes the barycentric subdivision of D and removes its minimal element in order to obtain the colorful system of parameters.

Ways to use colorfulSOP :

colorfulSOP(SimplicialComplex)
colorfulSOP(SimplicialComplex,ZZ)

For the programmer

The object colorfulSOP is a method function.