isWeaklyGVD -- checks whether an ideal is weakly geometrically vertex decomposable

This function tests whether an ideal $I \subseteq k[x_1,\ldots,x_n]$ is weakly geometrically vertex decomposable [KR, Definition 4.6].

See isGVD for the definition of the ideals $C_{y,I}$ and $N_{y,I}$ used below. We say that a geometric vertex decomposition is degenerate if $C_{y,I} = \langle 1 \rangle$ or if $\sqrt{C_{y,I}} = \sqrt{N_{y,I}}$. The geometric vertex decomposition is nondegenerate otherwise.

An ideal $I \subseteq R = k[x_1, \ldots, x_n]$ is weakly geometrically vertex decomposable if $I$ is unmixed and

(1) $I = \langle 1 \rangle$, or $I$ is generated by a (possibly empty) subset of variables of $R$, or

(2) (Degenerate Case) for some variable $y = x_j$ of $R$, ${\rm in}_y(I) = C_{y,I} \cap (N_{y,I} + \langle y \rangle)$ is a degenerate geometric vertex decomposition and the contraction of $N_{y,I}$ to the ring $k[x_1,\ldots,\hat{y},\ldots,x_n]$ is weakly geometrically vertex decomposable, or

(3) (Nondegenerate Case) for some variable $y = x_j$ of $R$, ${\rm in}_y(I) = C_{y,I} \cap (N_{y,I} + \langle y \rangle)$ is a nondegenerate geometric vertex decomposition, the contraction of $C_{y,I}$ to the ring $k[x_1,\ldots,\hat{y},\ldots,x_n]$ is weakly geometrically vertex decomposable, and $N_{y,I}$ is radical and Cohen-Macaulay.

The following example is [KR, Example 4.10]. It is an example of an ideal that is weakly geometrically vertex decomposable, but not geometrically vertex decomposable.

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

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

o2 : Ideal of R

i3 : isWeaklyGVD I

o3 = true

i4 : isGVD I

o4 = false

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