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# 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 Sergio Da Silva, Megumi Harada, Patricia Klein, and Jenna Rajchgot for feedback and suggestions. Additionally, we thank the anonymous referees of the paper [CVT] for their concrete suggestions that significantly improved that manuscript and this package. Cummings was partially supported by an NSERC USRA and CGS-M and a Milos Novotny Fellowship. Van Tuyl's research is partially supported by NSERC Discovery Grant 2019-05412.

## References

[CDSRVT] Mike Cummings, Sergio Da Silva, Jenna Rajchgot, and Adam Van Tuyl. Geometric vertex decomposition and liaison for toric ideals of graphs. Algebr. Comb., 6 (2023), no. 4, 965–997.

[CVT] Mike Cummings and Adam Van Tuyl. The GeometricDecomposability package for Macaulay2. J. Softw. Algebra Geom., 14 (2024), no. 1, 41–50.

[DSH] Sergio Da Silva and Megumi Harada. Geometric vertex decomposition, Gröbner bases, and Frobenius splittings for regular nilpotent Hessenberg Varieties. Transform. Groups, 2023.

[KMY] Allen Knutson, Ezra Miller, and Alexander Yong. Gröbner geometry of vertex decompositions and of flagged tableaux. J. Reine Angew. Math. 630 (2009), 1–-31.

[KR] Patricia Klein and Jenna Rajchgot. Geometric vertex decomposition and liaison. Forum Math. Sigma, 9 (2021) Paper No. e70, 23pp.

[SM] Hero Saremi and Amir Mafi. Unmixedness and arithmetic properties of matroidal ideals. Arch. Math. 114 (2020), no. 3 299–-304.

### Menu

- oneStepGVDNyI -- computes the ideal $N_{y,I}$
- 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
- initialYForms -- computes the ideal of initial y-forms
- 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
- oneStepGVDCyI -- computes the ideal $C_{y,I}$
- oneStepGVD -- computes a geometric vertex decomposition
- 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
- IsIdealHomogeneous -- specify whether an ideal is homogeneous
- IsIdealUnmixed -- specify whether an ideal is unmixed
- OnlyDegenerate -- restrict to degenerate geometric vertex decompositions
- OnlyNondegenerate -- restrict to nondegenerate geometric vertex decompositions
- SquarefreeOnly -- only return the squarefree variables from the generators
- UniversalGB -- whether the generators for an ideal form a universal Gröbner basis

## Authors

- Mike Cummings <mike.cummings@uwaterloo.ca>
- Adam Van Tuyl <vantuyl@math.mcmaster.ca>

## Certification

Version 1.2 of this package was accepted for publication in volume 14 of Journal of Software for Algebra and Geometry on 2024-01-23, in the article The GeometricDecomposability package for Macaulay2 (DOI: 10.2140/jsag.2024.14.41). That version can be obtained from the journal.

## Version

This documentation describes version 1.4.3 of GeometricDecomposability, released June 10, 2025.

## Citation

If you have used this package in your research, please cite it as follows:

    @misc{GeometricDecomposabilitySource,
      title = {{GeometricDecomposability: checking whether ideals are geometrically vertex decomposable. Version~1.4.3}},
      author = {Mike Cummings and Adam Van Tuyl},
      howpublished = {A \emph{Macaulay2} package available at
        \url{https://github.com/Macaulay2/M2/tree/stable/M2/Macaulay2/packages}}
    }

    @article{GeometricDecomposabilityArticle,
      title = {{The GeometricDecomposability package for \emph{Macaulay2}}},
      author = {Mike Cummings and Adam Van Tuyl},
      journal = {Journal of Software for Algebra and Geometry},
      volume = {14},
      year = {2024},
    }

## Exports

- Functions and commands
  - 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
  - initialYForms -- computes the ideal of initial y-forms
  - 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
  - oneStepGVD -- computes a geometric vertex decomposition
  - oneStepGVDCyI -- computes the ideal $C_{y,I}$
  - oneStepGVDNyI -- computes the ideal $N_{y,I}$
- Methods
  - `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
  - `initialYForms(Ideal,RingElement)` -- see initialYForms -- computes the ideal of initial y-forms
  - `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
  - `oneStepGVD(Ideal,RingElement)` -- see oneStepGVD -- computes a geometric vertex decomposition
  - `oneStepGVDCyI(Ideal,RingElement)` -- see oneStepGVDCyI -- computes the ideal $C_{y,I}$
  - `oneStepGVDNyI(Ideal,RingElement)` -- see oneStepGVDNyI -- computes the ideal $N_{y,I}$
- Symbols
  - 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
  - IsIdealHomogeneous -- specify whether an ideal is homogeneous
  - IsIdealUnmixed -- specify whether an ideal is unmixed
  - OnlyDegenerate -- restrict to degenerate geometric vertex decompositions
  - OnlyNondegenerate -- restrict to nondegenerate geometric vertex decompositions
  - SquarefreeOnly -- only return the squarefree variables from the generators
  - UniversalGB -- whether the generators for an ideal form a universal Gröbner basis

## For the programmer

The object GeometricDecomposability is a package, defined in GeometricDecomposability.m2.

The source of this document is in /build/macaulay2-88fgJW/macaulay2-1.25.11+ds/M2/Macaulay2/packages/GeometricDecomposability.m2:897:0.