isGVD I
This function tests whether a given ideal is geometrically vertex decomposable. Geometrically vertex decomposable ideals are based upon the geometric vertex decomposition defined by Knutson, Miller, and Yong [KMY]. Using geometric vertex decomposition, Klein and Rajchgot gave a recursive definition for geometrically vertex decomposable ideals in [KR, Definition 2.7]. This definition generalizes the properties of a squarefree monomial ideal whose associated simplicial complex is vertex decomposable.
We include the definition here. Let $y$ be a variable of the polynomial ring $R = k[x_1,\ldots,x_n]$. A monomial ordering $<$ on $R$ is said to be $y$compatible if the initial term of $f$ satisfies ${\rm in}_<(f) = {\rm in}_<({\rm in}_y(f))$ for all $f \in R$. Here, ${\rm in}_y(f)$ is the initial $y$form of $f$, that is, if $f = \sum_i \alpha_iy^i$ and $\alpha_d \neq 0$ but $\alpha_t = 0$ for all $t >d$, then ${\rm in}_y(f) = \alpha_d y^d$. We set ${\rm in}_y(I) = \langle {\rm in}_y(f) ~~ f \in I \rangle$ to be the ideal generated by all the initial $y$forms in $I$.
Given an ideal $I$ and a $y$compatible monomial ordering $<$, let $G(I) = \{ g_1,\ldots,g_m\}$ be a Gröbner basis of $I$ with respect to this ordering. For $i=1,\ldots,m$, write $g_i$ as $g_i = y^{d_i}q_i + r_i$, where $y$ does not divide any term of $q_i$; that is, ${\rm in}_y(g_i) = y^{d_i}q_i$. Given this setup, we define two ideals: $$C_{y,I} = \langle q_1,\ldots,q_m\rangle$$ and $$N_{y,I} = \langle q_i ~~ d_i = 0 \rangle.$$ Recall that an ideal $I$ is unmixed if all of the associated primes of $I$ have the same height.
An ideal $I$ of $R =k[x_1,\ldots,x_n]$ is 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) there is a variable $y = x_i$ in $R$ and a $y$compatible monomial ordering $<$ such that $${\rm in}_y(I) = C_{y,I} \cap (N_{y,I} + \langle y \rangle),$$ and the contractions of the ideals $C_{y,I}$ and $N_{y,I}$ to the ring $k[x_1,\ldots,\hat{y},\ldots,x_n]$ are geometrically vertex decomposable.
NOTE: The ideals $C_{y,I}$ and $N_{y,I}$ do not depend upon the choice of the Gröbner basis or a particular $y$compatible order (see comment after [KR, Definition 2.3]). When computing $C_{y,I}$ and $N_{y,I}$ we use a lexicographical ordering on $R$ where $y > x_j$ for all $i \neq j$ if $y = x_i$ since this gives us a $y$compatible order.




Squarefree monomial ideals that are geometrically vertex decomposable are precisely those squarefree monomial ideals whose associated simplicial complex are vertex decomposable [KR, Proposition 2.9]. The edge ideal of a chordal graph corresponds to a simplicial complex that is vertex decomposable (for more, see the EdgeIdeals package). The option Verbose shows the intermediate steps; in particular, Verbose displays what variable is being used to test a decomposition, as well as the ideals $C_{y,I}$ and $N_{y,I}$.



The following is an example of a toric ideal of graph that is geometrically vertex decomposable, and another example of a toric ideal of a graph that is not geometrically vertex decomposable. The second ideal is not CohenMacaulay, so it cannot be geometrically vertex decomposable [KR, Corollary 4.5]. For background on toric ideals of graphs, see [CDSRVT, Section 3].






[CDSRVT] Mike Cummings, Sergio Da Silva, Jenna Rajchgot, and Adam Van Tuyl. Geometric vertex decomposition and liaison for toric ideals of graphs. To appear in Algebraic Combinatorics, preprint available at arXiv:2207.06391 (2022).
[KMY] Allen Knutson, Ezra Miller, and Alexander Yong. Gröbner geometry of vertex decompositions and of flagged tableaux. J. Reine Angew. Math. 630 (2009) 1–31.
[KR] Patricia Klein and Jenna Rajchgot. Geometric vertex decomposition and liaison. Forum Math. Sigma, 9 (2021) e70:123.
The object isGVD is a method function with options.