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
- 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
- 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
- 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
- oneStepGVD -- computes a geometric vertex decomposition
- oneStepGVDCyI -- computes the ideal $C_{y,I}$ for a given ideal and indeterminate
- oneStepGVDNyI -- computes the ideal $N_{y,I}$ for a given ideal and indeterminate
- 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
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 1.4.2 of GeometricDecomposability.
Citation
If you have used this package in your research, please cite it as follows:
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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 $C_{y,I}$ for a given ideal and indeterminate
- oneStepGVDNyI -- computes the ideal $N_{y,I}$ for a given ideal and indeterminate
- 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 $C_{y,I}$ for a given ideal and indeterminate
- oneStepGVDNyI(Ideal,RingElement) -- see oneStepGVDNyI -- computes the ideal $N_{y,I}$ for a given ideal and indeterminate
- 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 GeometricDecomposability.m2:895:0.