isLexCompatiblyGVD -- checks whether an ideal is <-compatibly geometrically vertex decomposable for a given order

Usage:

isLexCompatiblyGVD(I, L)
Inputs:
- I, an ideal,
- L, a list,
Optional inputs:
- CheckCM => ..., default value "always", when to perform a Cohen-Macaulay check on the ideal
- CheckUnmixed => ..., default value true, check whether ideals encountered are unmixed
- IsIdealHomogeneous => ..., default value false, specify whether an ideal is homogeneous
- IsIdealUnmixed => ..., default value false, specify whether an ideal is unmixed
- UniversalGB => ..., default value false, whether the generators for an ideal form a universal Gröbner basis
- Verbose => ..., default value false
Outputs:
- a Boolean value,

Description

An ideal $I$ is $<$-compatibly geometrically vertex decomposable if there exists a (lexicographic) order $<$ such that $I$ is geometrically vertex decomposable and for every (one-step) geometric vertex decomposition, we pick $y$ to be the most expensive indeterminate remaining in the ideal according to $<$ [KR, Definition 2.11]. For the definition of a (one-step) geometric vertex decomposition, see oneStepGVD.

This method returns a Boolean value depending upon whether or not the given ideal is $<$-compatibly geometrically vertex decomposable with respect to a given ordering lex ordering of the indeterminates. Compare this function to the command findLexCompatiblyGVDOrders which checks all possible lex orders of the variables in order to find at least one $<$-compatibly lex order.

Below is [KR, Example 2.16], which is an example of an ideal that is not $<$-compatibly geometrically vertex decomposable. Any permutation of the variables we give in this example will result in false.

i1 : R = QQ[x,y,z,w,r,s];

i2 : I = ideal(y*(z*s - x^2), y*w*r, w*r*(z^2 + z*x + w*r + s^2));

o2 : Ideal of R

i3 : isLexCompatiblyGVD(I, {x,y,z,w,r,s})

o3 = false

i4 : isLexCompatiblyGVD(I, {s,x,w,y,r,z}, Verbose=>true)
I = ideal(-x^2*y+y*z*s,y*w*r,x*z*w*r+z^2*w*r+w^2*r^2+w*r*s^2)
initial ideal = ideal(w*y*r,s*y*z,s^2*w*r)
I = ideal(w*y*r,s*y*z,s^2*w*r)
-- decomposing with respect to s
Warning: not a valid geometric vertex decomposition

o4 = false

In view of [KR, Proposition 2.14], we check whether the initial ideal ${\rm in}_<(I)$ is $<$-compatibly geometrically vertex decomposable. Heuristically, one should expect this check to be quicker than the definition [KR, Definition 2.11] since checking unmixedness (that is, computing a primary decomposition) is in general faster for monomial ideals.

References

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

Ways to use isLexCompatiblyGVD:

isLexCompatiblyGVD(Ideal,List)

For the programmer

The object isLexCompatiblyGVD is a method function with options.

The source of this document is in GeometricDecomposability.m2:1326:0.

isLexCompatiblyGVD -- checks whether an ideal is <-compatibly geometrically vertex decomposable for a given order

Description

References

See also

Ways to use isLexCompatiblyGVD:

For the programmer