Adam Van Tuyl
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View article: Splittings of Ideals of Points in $\mathbb{P}^{1}\times\mathbb{P}^{1}$
Splittings of Ideals of Points in $\mathbb{P}^{1}\times\mathbb{P}^{1}$ Open
Let $I_\mathbb{X}$ be the bihomogeneous ideal of a finite set of points $\mathbb{X} \subseteq \mathbb{P}^1 \times \mathbb{P}^1$. The purpose of this note is to consider ``splittings'' of the ideal $I_\mathbb{X}$, that is, finding ideals $J…
View article: Analytic spread of binomial edge ideals
Analytic spread of binomial edge ideals Open
We investigate the analytic spread of binomial edge ideals of finite simple graphs. We provide tight bounds for this invariant in general. For special families of graphs (e.g., closed graphs, pseudo-forests), we compute the exact value for…
View article: Sign patterns which require or allow the strong multiplicity property
Sign patterns which require or allow the strong multiplicity property Open
We initiate a study of sign patterns that require or allow the non-symmetric strong multiplicity property (nSMP). We show that all cycle patterns require the nSMP, regardless of the number of nonzero diagonal entries. We present a class of…
View article: Levelable graphs
Levelable graphs Open
We study a family of positive weighted well-covered graphs, which we call levelable graphs, that are related to a construction of level artinian rings in commutative algebra. A graph $G$ is levelable if there exists a weight function with …
View article: A classification of van der Waerden complexes with linear resolution
A classification of van der Waerden complexes with linear resolution Open
In 2017, Ehrenborg, Govindaiah, Park, and Readdy defined the van der Waerden complex ${\tt vdW}(n,k)$ to be the simplicial complex whose facets correspond to all the arithmetic sequences on the set $\{1,\ldots,n\}$ of a fixed length $k$. T…
View article: Partial Betti splittings with applications to binomial edge ideals
Partial Betti splittings with applications to binomial edge ideals Open
We introduce the notion of a partial Betti splitting of a homogeneous ideal, generalizing the notion of a Betti splitting first given by Francisco, Hà, and Van Tuyl. Given a homogeneous ideal $I$ and two ideals $J$ and $K$ such that $I = J…
View article: Simplicial Complexes with Many Facets are Vertex Decomposable
Simplicial Complexes with Many Facets are Vertex Decomposable Open
Suppose $\Delta$ is a pure simplicial complex on $n$ vertices having dimension $d$ and let $c = n-d-1$ be its codimension in the simplex. Terai and Yoshida proved that if the number of facets of $\Delta$ is at least $\binom{n}{c}-2c+1$, th…
View article: The GeometricDecomposability package for Macaulay2
The GeometricDecomposability package for Macaulay2 Open
1. INTRODUCTION.The geometric vertex decomposition of an ideal was first introduced by Knutson, Miller, and Yong [9] as part of their study of vexillary matrix Schubert varieties.Geometric vertex decomposition can be viewed as a generaliza…
View article: Simplicial complexes with many facets are vertex decomposable
Simplicial complexes with many facets are vertex decomposable Open
Suppose $Δ$ is a pure simplicial complex on $n$ vertices having dimension $d$ and let $c = n-d-1$ be its codimension in the simplex. Terai and Yoshida proved that if the number of facets of $Δ$ is at least $\binom{n}{c}-2c+1$, then $Δ$ is …
View article: Conditions for Virtually Cohen--Macaulay Simplicial Complexes
Conditions for Virtually Cohen--Macaulay Simplicial Complexes Open
A simplicial complex $Δ$ is a virtually Cohen-Macaulay simplicial complex if its associated Stanley-Reisner ring $S$ has a virtual resolution, as defined by Berkesch, Erman, and Smith, of length ${\rm codim}(S)$. We provide a sufficient co…
View article: Hadamard products and binomial ideals
Hadamard products and binomial ideals Open
We study the Hadamard product of two varieties V and W, with particular attention to the situation when one or both of V and W is a binomial variety. The main result of this paper shows that when V and W are both binomial varieties, and th…
View article: Three invariants of geometrically vertex decomposable ideals
Three invariants of geometrically vertex decomposable ideals Open
We study three invariants of geometrically vertex decomposable ideals: the Castelnuovo-Mumford regularity, the multiplicity, and the $a$-invariant. We show that these invariants can be computed recursively using the ideals that appear in t…
View article: Fröberg's Theorem, vertex splittability and higher independence complexes
Fröberg's Theorem, vertex splittability and higher independence complexes Open
A celebrated theorem of Fröberg gives a complete combinatorial classification of quadratic square-free monomial ideals with a linear resolution. A generalization of this theorem to higher degree square-free monomial ideals is an active are…
View article: Geometric vertex decomposition and liaison for toric ideals of graphs
Geometric vertex decomposition and liaison for toric ideals of graphs Open
Geometric vertex decomposability for polynomial ideals is an ideal-theoretic generalization of vertex decomposability for simplicial complexes. Indeed, a homogeneous geometrically vertex decomposable ideal is radical and Cohen-Macaulay, an…
View article: Comparing invariants of toric ideals of bipartite graphs
Comparing invariants of toric ideals of bipartite graphs Open
Let be a finite simple graph and let denote its associated toric ideal in the polynomial ring . For each integer , we completely determine all the possible values for the tuple when is a connected bipartite graph on vertices.
View article: The weak Lefschetz property of whiskered graphs
The weak Lefschetz property of whiskered graphs Open
We consider Artinian level algebras arising from the whiskering of a graph. Employing a result by Dao-Nair we show that multiplication by a general linear form has maximal rank in degrees 1 and $n-1$ when the characteristic is not two, whe…
View article: Down-left graphs and a connection to toric ideals of graphs
Down-left graphs and a connection to toric ideals of graphs Open
We introduce a family of graphs, which we call down-left graphs, and study their combinatorial and algebraic properties. We show that members of this family are well-covered, $C_5$-free, and vertex decomposable. By applying a result of Hà-…
View article: Comparing invariants of toric ideals of bipartite graphs
Comparing invariants of toric ideals of bipartite graphs Open
Let $G$ be a finite simple graph and let $I_G$ denote its associated toric ideal in the polynomial ring $R$. For each integer $n\geq 2$, we completely determine all the possible values for the tuple $({\rm reg}(R/I_G), {\rm deg}(h_{R/I_G}(…
View article: Condition Numbers of Hessenberg Companion Matrices
Condition Numbers of Hessenberg Companion Matrices Open
The Fiedler matrices are a large class of companion matrices that include the well-known Frobenius companion matrix. The Fiedler matrices are part of a larger class of companion matrices that can be characterized with a Hessenberg form. In…
View article: Hadamard Products and Binomial Ideals
Hadamard Products and Binomial Ideals Open
We study the Hadamard product of two varieties $V$ and $W$, with particular attention to the situation when one or both of $V$ and $W$ is a binomial variety. The main result of this paper shows that when $V$ and $W$ are both binomial varie…
View article: The GeometricDecomposability package for Macaulay2
The GeometricDecomposability package for Macaulay2 Open
Using the geometric vertex decomposition property first defined by Knutson, Miller, and Yong, a recursive definition for geometrically vertex decomposable ideals was given by Klein and Rajchgot. We introduce the Macaulay2 package Geometric…
View article: Geometric vertex decomposition and liaison for toric ideals of graphs
Geometric vertex decomposition and liaison for toric ideals of graphs Open
The geometric vertex decomposability property for polynomial ideals is an ideal-theoretic generalization of the vertex decomposability property for simplicial complexes. Indeed, a homogeneous geometrically vertex decomposable ideal is radi…
View article: The regularity and h-polynomial of Cameron-Walker graphs
The regularity and h-polynomial of Cameron-Walker graphs Open
Fix an integer n ≥ 1, and consider the set of all connected finite simple graphs on n vertices.For each G in this set, let I(G) denote the edge ideal of G in the polynomial ring R = K[x 1 , . . ., x n ].We initiate a study of the set RD(n)…
View article: Powers of componentwise linear ideals: The Herzog--Hibi--Ohsugi Conjecture and related problems
Powers of componentwise linear ideals: The Herzog--Hibi--Ohsugi Conjecture and related problems Open
In 1999 Herzog and Hibi introduced componentwise linear ideals. A homogeneous ideal $I$ is componentwise linear if for all non-negative integers $d$, the ideal generated by the homogeneous elements of degree $d$ in $I$ has a linear resolut…
View article: Virtual resolutions of points in $\mathbb{P}^1 \times \mathbb{P}^1$
Virtual resolutions of points in $\mathbb{P}^1 \times \mathbb{P}^1$ Open
We explore explicit virtual resolutions, as introduced by Berkesch, Erman, and Smith, for ideals of sets of points in $\mathbb{P}^1 \times \mathbb{P}^1$. Specifically, we describe a virtual resolution for a sufficiently general set of poin…
View article: On the Waldschmidt constant of square-free principal Borel ideals
On the Waldschmidt constant of square-free principal Borel ideals Open
Fix a square-free monomial $m \in S = \mathbb{K}[x_1,\ldots,x_n]$. The square-free principal Borel ideal generated by $m$, denoted ${\rm sfBorel}(m)$, is the ideal generated by all the square-free monomials that can be obtained via Borel m…