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

## Synopsis

• Usage:
findLexCompatiblyGVDOrders I
• Inputs:
• Optional inputs:
• CheckUnmixed => ..., default value true, check whether ideals encountered are unmixed
• Outputs:
• a list, if no order exists, returns {}, otherwise returns L, a list containing all the lexicographical orders which work

## 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] P. Klein and J. Rajchgot. Geometric Vertex Decomposition and Liaison. Forum of 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.