findOneStepGVD I
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.
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The following example is [KR, Example 2.16]. The variable $b$ is the only indeterminate for which there exists a geometric vertex decomposition.
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[KR] Patricia Klein and Jenna Rajchgot. Geometric vertex decomposition and liaison. Forum Math. Sigma, 9 (2021) e70:1-23.
The object findOneStepGVD is a method function with options.