findLexCompatiblyGVDOrders -- finds all lexicographic monomial orders $<$ such that the ideal is $<$-compatibly geometrically vertex decomposable

Synopsis

Usage:

findLexCompatiblyGVDOrders I
Inputs:
- I, an ideal,
Optional inputs:
- CheckUnmixed => ..., default value true, check whether ideals encountered are unmixed
Outputs:
- a list, list containing all the lexicographical orders $<$ with respect to which I is $<$-compatibly geometrically vertex decomposable

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 computes all possible lex orders $<$ for which the ideal $I$ is $<$-compatibly geometrically vertex decomposable.

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

i2 : I = ideal(x-y, x-z);

o2 : Ideal of R

i3 : findLexCompatiblyGVDOrders I

o3 = {{x, y, z}, {x, z, y}, {y, x, z}, {y, z, x}, {z, x, y}, {z, y, x}}

o3 : List

The ideal in the following example is not square-free with respect to any indeterminate, so no one-step geometric vertex decomposition exists.

i4 : R = QQ[x,y];

i5 : I = ideal(x^2-y^2);

o5 : Ideal of R

i6 : findLexCompatiblyGVDOrders I

o6 = {}

o6 : List

References

[KR] Patricia Klein and Jenna Rajchgot. Geometric vertex decomposition and liaison. Forum Math. Sigma, 9 (2021) e70:1-23.

Caveat

In the ring $k[x_1, \ldots, x_n]$, there are $n!$ possible lexicographic monomial orders, so this function can be computationally expensive.

Ways to use findLexCompatiblyGVDOrders:

findLexCompatiblyGVDOrders(Ideal)

For the programmer

The object findLexCompatiblyGVDOrders is a method function with options.