This package includes routines to check whether an ideal is geometrically vertex decomposable.
Geometrically vertex decomposable ideals can be viewed as a generalization of the properties of the Stanley-Reisner ideal of a vertex decomposable simplicial complex. This family of ideals is based upon the geometric vertex decomposition property defined by Knutson, Miller, and Yong [KMY]. Klein and Rajchgot then gave a recursive definition for geometrically vertex decomposable ideals in [KR] using this notion.
An unmixed ideal $I$ in a polynomial ring $R$ is geometrically vertex decomposable if it is the zero ideal, the unit ideal, an ideal generated by indeterminates, or if there is a indeterminate $y$ of $R$ such that two ideals $C_{y,I}$ and $N_{y,I}$ constructed from $I$ are both geometrically vertex decomposable. For the complete definition, see isGVD.
Observe that a geometrically vertex decomposable ideal is recursively defined. The complexity of verifying that an ideal is geometrically vertex decomposable will increase as the number of indeterminates appearing in the ideal increases.
We thank S. Da Silva, P. Klein, J. Rajchgot, and M. Harada for feedback. Cummings was partially supported by an NSERC USRA. Van Tuyl's research is partially supported by NSERC Discovery Grant 2019-05412.
[CDSRVT] M. Cummings, S. Da Silva, J. Rajchgot, and A. Van Tuyl. Geometric Vertex Decomposition and Liaison for Toric Ideals of Graphs. Preprint, arXiv:2207.06391 (2022).
[DSH] S. Da Silva and M. Harada. Regular Nilpotent Hessenberg Varieties, Gröbner Bases, and Toric Degenerations. Preprint, arXiv:2207.08573 (2022).
[KMY] A. Knutson, E. Miller, and A. Yong. Gröbner Geometry of Vertex Decompositions and of Flagged Tableaux. J. Reine Angew. Math. 630 (2009) 1–31.
[KR] P. Klein and J. Rajchgot. Geometric Vertex Decomposition and Liaison. Forum of Math, Sigma, 9 (2021) e70:1-23.
[SM] H. Saremi and A. Mafi. Unmixedness and Arithmetic Properties of Matroidal Ideals. Arch. Math. 114 (2020) 299–304.
This documentation describes version 1.2 of GeometricDecomposability.
The source code from which this documentation is derived is in the file GeometricDecomposability.m2.
The object GeometricDecomposability is a package.