Project: Geometric Vertex Decomposition

• People involved: Mike Cummings (McMaster), Adam Van Tuyl (McMaster)
• Goal: Create a package for functions related to the geometrically vertex decomposability of an ideal
• Current status: Submitted by pull request to M2's master branch

Project Description

A geometric vertex decomposition of an ideal is a generalization of the properties of a Stanley-Reisner ideal of a vertex decomposable simplicial complex. First introduced by Klein and Rajchgot [2], geometric vertex decomposability provides a new tool to show that an ideal is in the Gorenstein liaison class of a complete intersection (glicci). Klein and Rajchgot's definition relies on the geometric vertex decomposition of an ideal first defined by Knutson, Miller, Yong [3].

The purpose of this package is to create code to test if a given homogeneous ideal is geometrically vertex decomposable.

Geometric vertex decomposability generalizes vertex decomposability for simplicial complexes. Vertex decomposability for simplicial complexes is already implemented for simplicial complexes in the SimplicialDecomposability package.

Examples

The following is an example of what we can currently do. The ideal i in the example below is the example of a toric ideal of a bipartite graph. In recent work, Cummings, Da Silva, Rajchgot, and Van Tuyl [1] have shown that these ideals are geometrically vertex decomposable. Our package verifies this. Note that the completed version will not show all the information we have presented here.

i1 : loadPackage "GeometricDecomposability"

o1 = GeometricDecomposability

o1 : Package

i2 : R = QQ[x_1..x_10]

o2 = R

o2 : PolynomialRing

i3 : i = ideal(x_1*x_2-x_3*x_4,x_4*x_5-x_6*x_7,x_7*x_8-x_9*x_10)

o3 = ideal (x x  - x x , x x  - x x , x x  - x x  )
             1 2    3 4   4 5    6 7   7 8    9 10

o3 : Ideal of R

i4 : isGVD i
ideal(x_1*x_2-x_3*x_4,x_4*x_5-x_6*x_7,x_7*x_8-x_9*x_10)
-- decomposing with respect to x_1
-- C = ideal(x_7*x_8-x_9*x_10,x_4*x_5-x_6*x_7,x_2)
-- N = ideal(x_7*x_8-x_9*x_10,x_4*x_5-x_6*x_7)
ideal(x_7*x_8-x_9*x_10,x_4*x_5-x_6*x_7,x_2)
-- decomposing with respect to x_2
-- C = ideal(x_7*x_8-x_9*x_10,x_4*x_5-x_6*x_7,1)
-- N = ideal(x_7*x_8-x_9*x_10,x_4*x_5-x_6*x_7)
ideal(x_7*x_8-x_9*x_10,x_4*x_5-x_6*x_7,1)
-- unit ideal
ideal(x_7*x_8-x_9*x_10,x_4*x_5-x_6*x_7)
-- decomposing with respect to x_4
-- C = ideal(x_7*x_8-x_9*x_10,x_5)
-- N = ideal(x_7*x_8-x_9*x_10)
ideal(x_7*x_8-x_9*x_10,x_5)
-- decomposing with respect to x_5
-- C = ideal(x_7*x_8-x_9*x_10,1)
-- N = ideal(x_7*x_8-x_9*x_10)
ideal(x_7*x_8-x_9*x_10,1)
-- unit ideal
ideal(x_7*x_8-x_9*x_10)
-- decomposing with respect to x_7
-- C = ideal x_8
-- N = ideal()
ideal x_8
-- generated by indeterminates
ideal()
-- zero ideal
ideal(x_7*x_8-x_9*x_10)
-- decomposing with respect to x_7
-- C = ideal x_8
-- N = ideal()
ideal x_8
-- generated by indeterminates
ideal()
-- zero ideal
ideal(x_7*x_8-x_9*x_10,x_4*x_5-x_6*x_7)
-- decomposing with respect to x_4
-- C = ideal(x_7*x_8-x_9*x_10,x_5)
-- N = ideal(x_7*x_8-x_9*x_10)
ideal(x_7*x_8-x_9*x_10,x_5)
-- decomposing with respect to x_5
-- C = ideal(x_7*x_8-x_9*x_10,1)
-- N = ideal(x_7*x_8-x_9*x_10)
ideal(x_7*x_8-x_9*x_10,1)
-- unit ideal
ideal(x_7*x_8-x_9*x_10)
-- decomposing with respect to x_7
-- C = ideal x_8
-- N = ideal()
ideal x_8
-- generated by indeterminates
ideal()
-- zero ideal
ideal(x_7*x_8-x_9*x_10)
-- decomposing with respect to x_7
-- C = ideal x_8
-- N = ideal()
ideal x_8
-- generated by indeterminates
ideal()
-- zero ideal

o4 = true

References (that describe the math and/or algorithms)

The majority of our package is built around the results in the following papers:

1. M. Cummings, S. Da Silva, J. Rajchgot, A. Van Tuyl, Geometric vertex decomposition and liaison for toric ideals of graphs. Preprint (2022).

2. P. Klein, J. Rajchgot, Geometric vertex decomposition and liaison. Forum of Mathematics, Sigma 9 (2021) Paper No. e70, 23 pp.

3. A. Knutson, E. Miller, A. Yong, Gröbner geometry of vertex decompositions and of flagged tableaux. Journal fur die Reine und Angewandte Mathematik 630 (2009) 1–31.