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

Synopsis

Usage:

isLexCompatiblyGVD(I, L)
Inputs:
- I, an ideal,
- L, a list,
Optional inputs:
- CheckCM => ..., default value "once", 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
- Verbose => ..., default value false, print additional output
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)
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)
-- decomposing with respect to s
Warning: Gröbner basis not square-free in s

o4 = false

References

[KR] P. Klein and J. Rajchgot. Geometric Vertex Decomposition and Liaison. Forum of Math, Sigma, 9 (2021) e70:1-23.

Ways to use `isLexCompatiblyGVD` :

"isLexCompatiblyGVD(Ideal,List)"

For the programmer

The object isLexCompatiblyGVD is a method function with options.