CheckDegenerate -- check whether the geometric vertex decomposition is degenerate

Description

A geometric vertex decomposition is degenerate if $\sqrt{C_{y,I}} = \sqrt{N_{y,I}}$ or if $C_{y,I} = \langle 1 \rangle$, and nondegenerate otherwise [KR, Section 2.2].

If CheckDegenerate=>true, then oneStepGVD returns a sequence of length four, where the fourth entry is either "degenerate" or "nondegenerate". Otherwise, oneStepGVD does not check whether the geometric vertex decomposition is degenerate and the sequence in the output has length three.

Note that the degeneracy of a geometric vertex decomposition does not matter with regards to whether an ideal is geometrically vertex decomposable. As a result, isGVD does not check this. However, the definition of weakly geometrically vertex decomposable depends the one-step geometric vertex decomposition at each step is degenerate, so isWeaklyGVD asks for this check.

 i1 : R = QQ[x,y,z] o1 = R o1 : PolynomialRing i2 : I = ideal(x-y, x-z) o2 = ideal (x - y, x - z) o2 : Ideal of R i3 : oneStepGVD(I, x, CheckDegenerate=>true) o3 = (true, ideal (y - z, 1), ideal(y - z), degenerate) o3 : Sequence

References

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