findOneStepGVD -- for which indeterminates does there exist a geometric vertex decomposition

Synopsis

Usage:

findOneStepGVD I
Inputs:
- I, an ideal,
Optional inputs:
- CheckUnmixed => ..., default value true, check whether ideals encountered are unmixed
- OnlyDegenerate => ..., default value false, restrict to degenerate geometric vertex decompositions
- OnlyNondegenerate => ..., default value false, restrict to nondegenerate geometric vertex decompositions
- SquarefreeOnly => ..., default value false, only return the squarefree variables from the generators
- UniversalGB => ..., default value false, whether the generators for an ideal form a universal Gröbner basis
- Verbose => ..., default value false
Outputs:
- a list,

Description

Returns a list containing the $y$ for which there exists a oneStepGVD. In other words, a list of all the variables $y$ that satisfy ${\rm in}_y(I) = C_{y,I} \cap (N_{y,I} + \langle y \rangle)$. All indeterminates $y$ which appear in the ideal are checked.

The results [KR, Lemma 2.6] and [KR, Lemma 2.12] are used to check whether $I$ has a geometric vertex decomposition with respect to each indeterminate $y$. First, for each indeterminate $y$ appearing in the ideal, we check whether the given generators of the ideal are squarefree in $y$. Note that this is a sufficient but not necessary condition. For the indeterminates $z$ that do not satisfy this sufficient condition, we compute a Gröbner of $I$ with respect to a $z$-compatible monomial order, and repeat the squarefree-check for the entries of this Gröbner basis.

Warning: if SquarefreeOnly=>true, then the options CheckUnmixed, OnlyDegenerate, and OnlyNondegenerate are ignored.

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 : findOneStepGVD I

o3 = {x, y, z}

o3 : List

The following example is [KR, Example 2.16]. The variable $b$ is the only indeterminate for which there exists a geometric vertex decomposition.

i4 : R = QQ[a..f]

o4 = R

o4 : PolynomialRing

i5 : I = ideal(b*(c*f - a^2), b*d*e, d*e*(c^2+a*c+d*e+f^2))

               2                             2       2 2        2
o5 = ideal (- a b + b*c*f, b*d*e, a*c*d*e + c d*e + d e  + d*e*f )

o5 : Ideal of R

i6 : findOneStepGVD I

o6 = {b}

o6 : List

References

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

Ways to use findOneStepGVD :

findOneStepGVD(Ideal)

For the programmer

The object findOneStepGVD is a method function with options.