GeometricDecomposability Changelog

This page describes changes by version for the Macaulay2 package GeometricDecomposability, by Mike Cummings and Adam Van Tuyl. The package and its documentation are available through the Macaulay2 website.

Version 1.5

We hope to include this update in version 1.26.11 of Macaulay2.

• Added an optional input AllowSub, implementing the notion of a geometric vertex decomposition allowing substitution.
• Changes to findLexCompatiblyGVDOrders:
  • Fixed a bug that would erroneously return the empty list
  • Introduced an optional parameter MaxNumTerms, default value infinity. When a finite (positive integer) value is specified, the code produces only produces candidate orderings of the variables of length at most MaxNumTerms.
    Note that this no longer guarantees that the orders in the output are sufficient to compatibly geometrically vertex decompose the given ideal. Instead, all that is guaranteed is that the given variables, in this order, produce geometric vertex decompositions, but makes no guarantee of geometric vertex decomposability.
    For example, running findLexCompatiblyGVDOrders(I, MaxNumTerms=>1) when I = ideal(x*y^2) returns {{x}}, but of course I is not (compatibly) geometrically vertex decomposable.
  • Introduced an optional parameter OnlyPrefix, default value true, which changes the default behaviour. This default behaviour returns prefixes of lex orderings that are sufficient to lex-compatibly geometrically vertex decompose the given ideal. Setting OnlyPrefix=>false recovers the previous behaviour, which returns all linear extensions of these prefixes.
• Renamed optional parameter SquarefreeOnly to OnlySquarefree

Version 1.4.3

Included in version 1.25.11 of Macaulay2.

• Added aliases link and deletion for oneStepGVDCyI and oneStepGVDNyI, respsectively.

Version 1.4.2

Included in version 1.25.05 of Macaulay2.

• Changed how isLexCompatiblyGVD is checked.

  Previously, we checked the definition directly. In this version, we now compute the initial ideal with respect to the given lex ordering < of the variables and check whether it is <-compatibly geometrically vertex decomposable. These are equivalent by Proposition 2.14 in Klein–Rajchgot's Geometric vertex decomposition and liaison.

  Heuristically we expect this new version to be faster because computing associated primes (and hence, checking unmixedness) is quicker for monomial ideals.

• Updated bibliography.

Version 1.4.1

Included in version 1.24 of Macaulay2.

• Fixed how geometric vertex decompositions are checked when a universal Gröbner basis is provided.

Version 1.4

Included in version 1.23 of Macaulay2.

• Added an optional input UniversalGB for when the given generating set of an ideal is a universal Gröbner basis. In this setting, the ideals C_y,I and N_y,I each inherit a universal Gröbner basis, and hence we can avoid computing Gröbner bases when checking whether I is geometrically vertex decomposable.
• Added an optional input SquarefreeOnly for the function findOneStepGVD. For an ideal I to have a geometric vertex decomposition with respect to y, it is sufficient for y to appear in the given generating set for I with maximum degree 1. When SquarefreeOnly=>true, we return only the variables that are squarefree in the generating set for I. This is much quicker, but does not necessarily provide an exhaustive list. The GVD methods now prefer these squarefree variables when computing C_y,I and N_y,I.
• Implemented suggestions from Patricia Klein:
  • Since Cohen-Macaulay implies unmixed, we need not always check unmixedness.
  • To check whether a homogeneous ideal is Cohen-Macaulay, we now check for equality between projective dimension and codimension. This is quicker than what it replaces, the isCM method from the Depth package, which verifies equality between dimension and depth. That these methods are equivalent in the homogeneous setting is guaranteed by Auslander–Buchsbaum.
  • The isGVD method now checks whether N_y,I is geometrically vertex decomposable before C_y,I. The link of a Cohen-Macaulay ideal will be Cohen-Macaulay, but the deletion need not be. So it may be more likely for N_y,I to not be Cohen-Macaulay and hence not geometrically vertex decomposable.

Version 1.3

• Renamed methods:
  • CyI has been renamed oneStepGVDCyI,
  • NyI has been renamed oneStepGVDNyI,
  • yInit has been renamed initialYForms.
• Updated isUnmixed; it should now run quicker

Version 1.2 ⭐

Included in version 1.22 of Macaulay2. This version was published in the Journal of Software for Algebra and Geometry.

• Fixed the error that would appear upon loading the package.
• Removed the documentation page for the optional input Verbose.

Version 1.1

First version of the package, included in version 1.21 of Macaulay2.