arXiv subjects

Adam Van Tuyl

Publications and source records attributed to Adam Van Tuyl.

At least 37 records · Page 2Linked to original sources

arXiv2023

Three invariants of geometrically vertex decomposable ideals

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 the geometric vertex decomposition process. As an application, we prove that the $a$-invariant of a geometrically vertex decomposable ideal is non-positive. We also recover some previously known results in the literature including a formula for the regularity of the Stanley--Reisner ideal of a pure vertex decomposable simplicial complex, and proofs that some well-known families of ideals are Hilbertian. Finally, we apply our recursions to the study of toric ideals of bipartite graphs. Included among our results on this topic is a new proof for a known bound on the $a$-invariant of a toric ideal of a bipartite graph.

math.AC↗

arXiv2023

The weak Lefschetz property of whiskered graphs

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, where $n$ is the number of vertices in the graph. Moreover, the multiplication is injective in degrees $<n/2$ when the characteristic is zero, following a proof by Hausel. Our result in the characteristic zero case is optimal in the sense that there are whiskered graphs for which the multiplication maps in all intermediate degrees $n/2,\ldots,n-2$ of the associated Artinian algebras fail to have maximal rank, and consequently, the weak Lefschetz property.

math.AC↗

arXiv2023

Down-left graphs and a connection to toric ideals of graphs

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-Woodroofe and Moradi--Khosh-Ahang, the (Castelnuovo-Mumford) regularity of the associated edge ideals is the induced matching number of the graph. As an application, we give a combinatorial interpretation for the regularity of the toric ideals of chordal bipartite graphs that are $(K_{3,3} \setminus e)$-free.

math.AC↗

arXiv2023

Comparing invariants of toric ideals of bipartite graphs

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 $({\rm reg}(R/I_G), {\rm deg}(h_{R/I_G}(t)),{\rm pdim}(R/I_G), {\rm depth}(R/I_G),\dim(R/I_G))$ when $G$ is a connected bipartite graph on $n$ vertices.

math.AC↗

arXiv2023

Condition Numbers of Hessenberg Companion Matrices

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 this paper, we demonstrate that the Hessenberg form of the Fiedler companion matrices provides a straight-forward way to compare the condition numbers of these matrices. We also show that there are other companion matrices which can provide a much smaller condition number than any Fiedler companion matrix. We finish by exploring the condition number of a class of matrices obtained from perturbing a Frobenius companion matrix while preserving the characteristic polynomial.

math.RA↗

arXiv2022

Hadamard Products and Binomial Ideals

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 the binomials that define $V$ and $W$ have the same binomial exponents, then the defining equations of $V \star W$ can be computed explicitly and directly from the defining equations of $V$ and $W$. This result recovers known results about Hadamard products of binomial hypersurfaces and toric varieties. Moreover, as an application of our main result, we describe a relationship between the Hadamard product of the toric ideal $I_G$ of a graph $G$ and the toric ideal $I_H$ of a subgraph $H$ of $G$. We also derive results about algebraic invariants of Hadamard products: assuming $V$ and $W$ are binomial with the same exponents, we show that $\text{deg}(V\star W) = \text{deg}(V)=\text{deg}(W)$ and $\dim(V\star W) = \dim(V)=\dim(W)$. Finally, given any (not necessarily binomial) projective variety $V$ and a point $p \in \mathbb{P}^n \setminus \mathbb{V}(x_0x_1\cdots x_n)$, subject to some additional minor hypotheses, we find an explicit binomial variety that describes all the points $q$ that satisfy $p \star V = q\star V$.

math.AC↗

arXiv2022

Virtual resolutions of points in $\mathbb{P}^1 \times \mathbb{P}^1$

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 points $X$ in $\mathbb{P}^1 \times \mathbb{P}^1$ that only depends on $|X|$. We also improve an existence result of Berkesch, Erman, and Smith in the special case of points in $\mathbb{P}^1 \times \mathbb{P}^1$; more precisely, we give an effective bound for their construction that gives a virtual resolution of length two for any set of points in $\mathbb{P}^1 \times \mathbb{P}^1$.

math.AC↗

arXiv2021

Powers of componentwise linear ideals: The Herzog--Hibi--Ohsugi Conjecture and related problems

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 resolution. For square-free monomial ideals, componentwise linearity is related via Alexander duality to the property of being sequentially Cohen-Macaulay for the corresponding simplicial complexes. In general, the property of being componentwise linear is not preserved by taking powers. In 2011, Herzog, Hibi, and Ohsugi conjectured that if $I$ is the cover ideal of a chordal graph, then $I^s$ is componentwise linear for all $s \geq 1$. We survey some of the basic properties of componentwise linear ideals, and then specialize to the progress on the Herzog-Hibi-Ohsugi conjecture during the last decade. We also survey the related problem of determining when the symbolic powers of a cover ideal are componentwise linear.

math.AC↗

arXiv2021

Powers of Principal $Q$-Borel ideals

Fix a poset $Q$ on $\{x_1,\ldots,x_n\}$. A $Q$-Borel monomial ideal $I \subseteq \mathbb{K}[x_1,\ldots,x_n]$ is a monomial ideal whose monomials are closed under the Borel-like moves induced by $Q$. A monomial ideal $I$ is a principal $Q$-Borel ideal, denoted $I=Q(m)$, if there is a monomial $m$ such that all the minimal generators of $I$ can be obtained via $Q$-Borel moves from $m$. In this paper we study powers of principal $Q$-Borel ideals. Among our results, we show that all powers of $Q(m)$ agree with their symbolic powers, and that the ideal $Q(m)$ satisfies the persistence property for associated primes. We also compute the analytic spread of $Q(m)$ in terms of the poset $Q$.

math.AC↗

arXiv2021

On the Waldschmidt constant of square-free principal Borel ideals

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 moves from the monomial $m$. We give upper and lower bounds for the Waldschmidt constant of ${\rm sfBorel}(m)$ in terms of the support of $m$, and in some cases, exact values. For any rational $\frac{a}{b} \geq 1$, we show that there exists a square-free principal Borel ideal with Waldschmidt constant equal to $\frac{a}{b}$.

math.AC↗

arXiv2021

Splittings of Toric Ideals

Let $I \subseteq R = \mathbb{K}[x_1,\ldots,x_n]$ be a toric ideal, i.e., a binomial prime ideal. We investigate when the ideal $I$ can be "split" into the sum of two smaller toric ideals. For a general toric ideal $I$, we give a sufficient condition for this splitting in terms of the integer matrix that defines $I$. When $I = I_G$ is the toric ideal of a finite simple graph $G$, we give additional splittings of $I_G$ related to subgraphs of $G$. When there exists a splitting $I = I_1+I_2$ of the toric ideal, we show that in some cases we can describe the (multi-)graded Betti numbers of $I$ in terms of the (multi-)graded Betti numbers of $I_1$ and $I_2$.

math.AC↗

arXiv2020

Well-covered Token Graphs

The $k$-token graph $T_k(G)$ is the graph whose vertices are the $k$-subsets of vertices of a graph $G$, with two vertices of $T_k(G)$ adjacent if their symmetric difference is an edge of $G$. We explore when $T_k(G)$ is a well-covered graph, that is, when all of its maximal independent sets have the same cardinality. For bipartite graphs $G$, we classify when $T_k(G)$ is well-covered. For an arbitrary graph $G$, we show that if $T_2(G)$ is well-covered, then the girth of $G$ is at most four. We include upper and lower bounds on the independence number of $T_k(G)$, and provide some families of well-covered token graphs.

math.CO↗

arXiv2020

Homological invariants of Cameron--Walker graphs

Let $G$ be a finite simple connected graph on $[n]$ and $R = K[x_1, \ldots, x_n]$ the polynomial ring in $n$ variables over a field $K$. The edge ideal of $G$ is the ideal $I(G)$ of $R$ which is generated by those monomials $x_ix_j$ for which $\{i, j\}$ is an edge of $G$. In the present paper, the possible tuples $(n, {\rm depth} (R/I(G)), {\rm reg} (R/I(G)), \dim R/I(G), {\rm deg} \ h(R/I(G)))$, where ${\rm deg} \ h(R/I(G))$ is the degree of the $h$-polynomial of $R/I(G)$, arising from Cameron--Walker graphs on $[n]$ will be completely determined.

math.AC↗

arXiv2020

Symbolic powers of codimension two Cohen-Macaulay ideals

Let $I_X$ be the saturated homogeneous ideal defining a codimension two arithmetically Cohen-Macaulay scheme $X \subseteq \mathbb{P}^n$, and let $I_X^{(m)}$ denote its $m$-th symbolic power. We are interested in when $I_X^{(m)} = I_X^m$. We survey what is known about this problem when $X$ is locally a complete intersection, and in particular, we review the classification of when $I_X^{(m)} = I_X^m$ for all $m \geq 1$. We then discuss how one might weaken these hypotheses, but still obtain equality between the symbolic and ordinary powers. Finally, we show that this classification allows one to: (1) simplify known results about symbolic powers of ideals of points in $\mathbb{P}^1 \times \mathbb{P}^1$; (2) verify a conjecture of Guardo, Harbourne, and Van Tuyl, and (3) provide additional evidence to a conjecture of Römer.

math.AC↗

arXiv2020

Regularity and h-polynomials of toric ideals of graphs

For all integers $4 \leq r \leq d$, we show that there exists a finite simple graph $G= G_{r,d}$ with toric ideal $I_G \subset R$ such that $R/I_G$ has (Castelnuovo-Mumford) regularity $r$ and $h$-polynomial of degree $d$. To achieve this goal, we identify a family of graphs such that the graded Betti numbers of the associated toric ideal agree with its initial ideal, and furthermore, this initial ideal has linear quotients. As a corollary, we can recover a result of Hibi, Higashitani, Kimura, and O'Keefe that compares the depth and dimension of toric ideals of graphs.

math.AC↗

arXiv2020

The regularity and $h$-polynomial of Cameron-Walker graphs

Fix an integer $n \geq 1$, 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 $R = K[x_1,\ldots,x_n]$. We initiate a study of the set $\mathcal{RD}(n) \subseteq \mathbb{N}^2$ consisting of all the pairs $(r,d)$ where $r = {\rm reg}(R/I(G))$, the Castelnuovo-Mumford regularity, and $d = {\rm deg} h_{R/I(G)}(t)$, the degree of the $h$-polynomial, as we vary over all the connected graphs on $n$ vertices. In particular, we identify sets $A(n)$ and $B(n)$ such that $A(n) \subseteq \mathcal{RD}(n) \subseteq B(n)$. When we restrict to the family of Cameron-Walker graphs on $n$ vertices, we can completely characterize all the possible $(r,d)$.

math.CO↗

arXiv2019

Partial coloring, vertex decomposability, and sequentially Cohen-Macaulay simplicial complexes

In attempting to understand how combinatorial modifications alter algebraic properties of monomial ideals, several authors have investigated the process of adding "whiskers" to graphs. In this paper, we study a similar construction to build a simplicial complex $Δ_χ$ from a coloring $χ$ of a subset of the vertices of $Δ$, and give necessary and sufficient conditions for this construction to produce vertex decomposable simplicial complexes. We apply this work to strengthen and give new proofs about sequentially Cohen-Macaulay edge ideals of graphs.

math.AC↗

arXiv2019

Hilbert functions of schemes of double and reduced points

It remains an open problem to classify the Hilbert functions of double points in $\mathbb{P}^2$. Given a valid Hilbert function $H$ of a zero-dimensional scheme in $\mathbb{P}^2$, we show how to construct a set of fat points $Z \subseteq \mathbb{P}^2$ of double and reduced points such that $H_Z$, the Hilbert function of $Z$, is the same as $H$. In other words, we show that any valid Hilbert function $H$ of a zero-dimensional scheme is the Hilbert function of a set of a positive number of double points and some reduced points. For some families of valid Hilbert functions, we are also able to show that $H$ is the Hilbert function of only double points. In addition, we give necessary and sufficient conditions for the Hilbert function of a scheme of a double points, or double points plus one additional reduced point, to be the Hilbert function of points with support on a star configuration of lines.

math.AC↗