# GeometricDecomposability -- a package to check whether ideals are geometrically vertex decomposable

## Description

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.

## Acknowledgement

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.

## References

[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.

## Version

This documentation describes version 1.1 of GeometricDecomposability.

## Source code

The source code from which this documentation is derived is in the file GeometricDecomposability.m2.

## Exports

• Functions and commands
• CyI -- computes the ideal $C_{y,I}$ for a given ideal and indeterminate
• findLexCompatiblyGVDOrders -- finds all lexicographic monomial orders $<$ such that the ideal is $<$-compatibly geometrically vertex decomposable
• findOneStepGVD -- for which indeterminates does there exist a geometric vertex decomposition
• getGVDIdeal -- computes the $C_{y,I}$ or $N_{y,I}$ ideal at any point in the GVD recursion tree
• isGeneratedByIndeterminates -- checks whether the ideal is generated by indeterminates
• isGVD -- checks whether an ideal is geometrically vertex decomposable
• isLexCompatiblyGVD -- checks whether an ideal is <-compatibly geometrically vertex decomposable for a given order
• isUnmixed -- checks whether an ideal is unmixed
• isWeaklyGVD -- checks whether an ideal is weakly geometrically vertex decomposable
• NyI -- computes the ideal $N_{y,I}$ for a given ideal and indeterminate
• oneStepGVD -- computes a geometric vertex decomposition
• yInit -- computes the ideal of initial y-forms
• Methods
• "CyI(Ideal,RingElement)" -- see CyI -- computes the ideal $C_{y,I}$ for a given ideal and indeterminate
• "findLexCompatiblyGVDOrders(Ideal)" -- see findLexCompatiblyGVDOrders -- finds all lexicographic monomial orders $<$ such that the ideal is $<$-compatibly geometrically vertex decomposable
• "findOneStepGVD(Ideal)" -- see findOneStepGVD -- for which indeterminates does there exist a geometric vertex decomposition
• "getGVDIdeal(Ideal,List)" -- see getGVDIdeal -- computes the $C_{y,I}$ or $N_{y,I}$ ideal at any point in the GVD recursion tree
• "isGeneratedByIndeterminates(Ideal)" -- see isGeneratedByIndeterminates -- checks whether the ideal is generated by indeterminates
• "isGVD(Ideal)" -- see isGVD -- checks whether an ideal is geometrically vertex decomposable
• "isLexCompatiblyGVD(Ideal,List)" -- see isLexCompatiblyGVD -- checks whether an ideal is <-compatibly geometrically vertex decomposable for a given order
• "isUnmixed(Ideal)" -- see isUnmixed -- checks whether an ideal is unmixed
• "isWeaklyGVD(Ideal)" -- see isWeaklyGVD -- checks whether an ideal is weakly geometrically vertex decomposable
• "NyI(Ideal,RingElement)" -- see NyI -- computes the ideal $N_{y,I}$ for a given ideal and indeterminate
• "oneStepGVD(Ideal,RingElement)" -- see oneStepGVD -- computes a geometric vertex decomposition
• "yInit(Ideal,RingElement)" -- see yInit -- computes the ideal of initial y-forms
• Symbols

## For the programmer

The object GeometricDecomposability is .

• CheckCM -- when to perform a Cohen-Macaulay check on the ideal
• CheckDegenerate -- check whether the geometric vertex decomposition is degenerate
• CheckUnmixed -- check whether ideals encountered are unmixed
• CyI -- computes the ideal $C_{y,I}$ for a given ideal and indeterminate
• findLexCompatiblyGVDOrders -- finds all lexicographic monomial orders $<$ such that the ideal is $<$-compatibly geometrically vertex decomposable
• findOneStepGVD -- for which indeterminates does there exist a geometric vertex decomposition
• getGVDIdeal -- computes the $C_{y,I}$ or $N_{y,I}$ ideal at any point in the GVD recursion tree
• isGeneratedByIndeterminates -- checks whether the ideal is generated by indeterminates
• isGVD -- checks whether an ideal is geometrically vertex decomposable
• IsIdealHomogeneous -- specify whether an ideal is homogeneous
• IsIdealUnmixed -- specify whether an ideal is unmixed
• isLexCompatiblyGVD -- checks whether an ideal is <-compatibly geometrically vertex decomposable for a given order
• isUnmixed -- checks whether an ideal is unmixed
• isWeaklyGVD -- checks whether an ideal is weakly geometrically vertex decomposable
• NyI -- computes the ideal $N_{y,I}$ for a given ideal and indeterminate
• oneStepGVD -- computes a geometric vertex decomposition
• OnlyDegenerate -- restrict to degenerate geometric vertex decompositions
• OnlyNondegenerate -- restrict to nondegenerate geometric vertex decompositions
• Verbose -- print additional output
• yInit -- computes the ideal of initial y-forms