Adam Van Tuyl

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

Huy Tai Ha, Adam Van Tuyl

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 hom…

math.AC2026

The van der Waerden Simplicial Complex and its Lefschetz Properties

Naveena Ragunathan, Adam Van Tuyl

The van der Waerden simplicial complex, denoted $vdw (n, k)$ , is the simpicial complex whose facets correspond to the arithmetic progressions of length $k$ in the set $\{1,\ldo…

Separators of fat points in P^n

Elena Guardo, Lucia Marino, Adam Van Tuyl

In this paper we extend the definition of a separator of a point P in P^n to a fat point P of multiplicity m. The key idea in our definition is to compare the fat point schemes Z =…

Conditions for Virtually Cohen--Macaulay Simplicial Complexes

Jay Yang, Adam Van Tuyl

A simplicial complex $I^$ is a virtually Cohen-Macaulay simplicial complex if its associated Stanley-Reisner ring $S$ has a virtual resolution, as defined by Berkesch, Erman, and S…

Betti numbers of toric ideals of graphs: A case study

Federico Galetto, Johannes Hofscheier, Graham Keiper +3

We compute the graded Betti numbers for the toric ideal of a family of graphs constructed by adjoining a cycle to a complete bipartite graph. The key observation is that this famil…

math.AC2001

The border of the Hilbert function of a set of points in P^{n_1} x ... x P^{n_k}

Adam Van Tuyl

We describe the eventual behaviour of the Hilbert function of a set of distinct points in P^{n_1} x ... x P^{n_k}. As a consequence of this result, we show that the Hilbert functio…

The $h$ -vectors of toric ideals of odd cycle compositions revisited

Kieran Bhaskara, Adam Van Tuyl, Sasha Zotine

Let $G$ be a graph consisting of $s$ odd cycles that all share a common vertex. Bhaskara, Higashitani, and Shibu Deepthi recently computed the $h$ -polynomial for the quotient ring…

math.CO2017

A note on the van der Waerden complex

Becky Hooper, Adam Van Tuyl

Ehrenborg, Govindaiah, Park, and Readdy recently introduced the van der Waerden complex, a pure simplicial complex whose facets correspond to arithmetic progressions. Using techniq…

A conjecture on critical graphs and connections to the persistence of associated primes

Christopher A. Francisco, Huy Tai Ha, Adam Van Tuyl

We introduce a conjecture about constructing critically (s+1)-chromatic graphs from critically s-chromatic graphs. We then show how this conjecture implies that any unmixed height…

math.CO2007

Shellable graphs and sequentially Cohen-Macaulay bipartite graphs

Adam Van Tuyl, Rafael H. Villarreal

Associated to a simple undirected graph G is a simplicial complex whose faces correspond to the independent sets of G. We call a graph G shellable if this simplicial complex is a s…

Algebraic properties of the path ideal of a tree

Jing Jane He, Adam Van Tuyl

The path ideal (of length t >=2) of a graph G is the monomial ideal, denoted I_t(G), whose generators correspond to the directed paths of length t in G. We study some of the algebr…

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

Jennifer Biermann, Beth Anne Castellano, Marcella Manivel +2

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, $C_{5}$ …

math.AC2026

Geometrically vertex decomposable star configurations

Susan M. Cooper, Emanuela Marangone, Elena Guardo +1

The goal of this paper is to determine how the family of ideals of star configurations intersects with the class of geometrically vertex decomposable ideals. The main result of thi…

Geometric vertex decomposition and liaison for toric ideals of graphs

Mike Cummings, Sergio Da Silva, Jenna Rajchgot +1

The geometric vertex decomposability property for polynomial ideals is an ideal-theoretic generalization of the vertex decomposability property for simplicial complexes. Indeed, a…

math.AG2012

Asymptotic resurgences for ideals of positive dimensional subschemes of projective space

Elena Guardo, Brian Harbourne, Adam Van Tuyl

Recent work of Ein-Lazarsfeld-Smith and Hochster-Huneke raised the problem of determining which symbolic powers of an ideal are contained in a given ordinary power of the ideal. Bo…

math.RA2008

Zero-nonzero patterns for nilpotent matrices over finite fields

Kevin N. Vander Meulen, Adam Van Tuyl

Fix a field F. A zero-nonzero pattern A is said to be potentially nilpotent over F if there exists a matrix with entries in F with zero-nonzero pattern A that allows nilpotence. In…

math.AC2004

On the defining ideal of a set of points in multi-projective space

Adam Van Tuyl

We investigate the defining ideal of a set of points X in multi-projective space with a special emphasis on the case that X is in generic position, that is, X has the maximal Hilbe…

The GeometricDecomposability package for Macaulay2

Mike Cummings, Adam Van Tuyl

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…

math.AG2018

Hadamard Star Configurations

Enrico Carlini, Maria Virginia Catalisano, Elena Guardo +1

Bocci, Carlini, and Kileel have shown that the square-free Hadamard product of a finite set of points $Z$ that all lie on a line $ℓ$ in $P^{n}$ produces a star configurat…

math.CO2026

Trivariate Splines on Fans of Hyperplane Arrangements and Koszul Homology

Carles Checa, Michael DiPasquale, Pablo MazÃ³n +5

We study the space of splines $S^{r} (I^£^{A})$ where $r$ denotes a smoothness distribution and $I^£^{A}$ is the fan of a central hyperpl…

The weak Lefschetz property of whiskered graphs

Susan M. Cooper, Sara Faridi, Thiago Holleben +2

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…

Colorings of hypergraphs, perfect graphs, and associated primes of powers of monomial ideals

Christopher A. Francisco, Huy Tai Ha, Adam Van Tuyl

There is a natural one-to-one correspondence between squarefree monomial ideals and finite simple hypergraphs via the cover ideal construction. Let H be a finite simple hypergraph,…

Symbolic powers versus regular powers of ideals of general points in P^1 x P^1

Elena Guardo, Brian Harbourne, Adam Van Tuyl

Recent work of Ein-Lazarsfeld-Smith and Hochster-Huneke raised the problem of which symbolic powers of an ideal are contained in a given ordinary power of the ideal. Bocci-Harbourn…

Sequentially Cohen-Macaulay Edge Ideals

Christopher A. Francisco, Adam Van Tuyl

Let G be a simple undirected graph on n vertices, and let I(G) \subseteq R = k[x_1,...,x_n] denote its associated edge ideal. We show that all chordal graphs G are sequentially Coh…

math.CO2018

Shedding vertices of vertex decomposable graphs

Jonathan Baker, Kevin N. Vander Meulen, Adam Van Tuyl

We focus our attention on well-covered graphs that are vertex decomposable. We show that for many known families of these vertex decomposable graphs, the set of shedding vertices f…

Separators of fat points in P^n x P^m

Elena Guardo, Adam Van Tuyl

We introduce definitions for the separator of a fat point and the degree of a fat point for a fat point scheme Z in P^n x P^m, and we study some of their properties.

math.CO2025

Spheres and balls as independence complexes

Susan M. Cooper, Sara Faridi, Thiago Holleben +2

The terms "whiskering", and more generally "grafting", refer to adding generators to any monomial ideal to make the resulting ideal Cohen-Macaulay. We investigate the independence…

math.CO2020

Well-covered Token Graphs

F. M. Abdelmalek, Esther Vander Meulen, Kevin N. Vander Meulen +1

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…

math.AG2012

Star configurations on generic hypersurfaces

Enrico Carlini, Elena Guardo, Adam Van Tuyl

Let $F$ be a homogeneous polynomial in $S = C [x_{0}, ..., x_{n}]$ . Our goal is to understand a particular polynomial decomposition of $F$ ; geometrically, we wish to determine wh…

Splittings of Ideals of Points in $P^{1} \times P^{1}$

Elena Guardo, Graham Keiper, Adam Van Tuyl

Let $I_{X}$ be the bihomogeneous ideal of a finite set of points $X \subseteq P^{1} \times P^{1}$ . The purpose of this note is to consider ``splitting…

math.AC2016

The Waldschmidt constant for squarefree monomial ideals

Cristiano Bocci, Susan Cooper, Elena Guardo +6

Given a squarefree monomial ideal $I \subseteq R = k [x_{1}, \dots, x_{n}]$ , we show that $I^\pm (I)$ , the Waldschmidt constant of $I$ , can be expressed as the optimal solution to a l…

math.AC2016

Bounds on the regularity of toric ideals of graphs

Jennifer Biermann, Augustine O'Keefe, Adam Van Tuyl

Let $G$ be a finite simple graph. We give a lower bound for the Castelnuovo-Mumford regularity of the toric ideal $I_{G}$ associated to $G$ in terms of the sizes and number of induce…

math.RA2010

Potentially Nilpotent Patterns and the Nilpotent-Jacobian Method

Hannah Bergsma, Kevin N. Vander Meulen, Adam Van Tuyl

A nonzero pattern is a matrix with entries in {0,*}. A pattern is potentially nilpotent if there is some nilpotent real matrix with nonzero entries in precisely the entries indicat…

ACM sets of points in multiprojective space

Elena Guardo, Adam Van Tuyl

If X is a finite set of points in a multiprojective space P^n1 x ... x P^nr with r >= 2, then X may or may not be arithmetically Cohen-Macaulay (ACM). For sets of points in P^1 x P…

math.AC2022

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

Megumi Harada, Maryam Nowroozi, Adam Van Tuyl

We explore explicit virtual resolutions, as introduced by Berkesch, Erman, and Smith, for ideals of sets of points in $P^{1} \times P^{1}$ . Specifically, we describe…

math.AC2002

Fat Points in P^1 x P^1 and their Hilbert Functions

Elena Guardo, Adam Van Tuyl

We study the Hilbert functions of fat points in P^1 x P^1. If Z is an arbitrary fat point subscheme of P^1 x P^1, then it can be shown that for every i and j the values of the Hilb…

On the Waldschmidt constant of square-free principal Borel ideals

Eduardo Camps Moreno, Craig Kohne, Eliseo Sarmiento +1

Fix a square-free monomial $m \in S = K [x_{1}, \dots, x_{n}]$ . The square-free principal Borel ideal generated by $m$ , denoted $sfBorel (m)$ , is the ideal generated by all…

On the linear strand of an edge ideal

Mike Roth, Adam Van Tuyl

Let I(G) be the edge ideal associated to a simple graph G. We study the graded Betti numbers that appear in the linear strand of the minimal free resolution of I(G).

math.RA2016

Refined Inertia of Matrix Patterns

Jonathan Earl, Kevin N. Vander Meulen, Adam Van Tuyl

We explore how the combinatorial arrangement of prescribed zeros in a matrix affects the possible eigenvalues that the matrix can obtain. We demonstrate that there are inertially a…

Splittable ideals and the resolutions of monomial ideals

Huy Tai Ha, Adam Van Tuyl

We provide a new combinatorial approach to study the minimal free resolutions of edge ideals, that is, quadratic square-free monomial ideals. With this method we can recover most o…

math.RA2025

Sign patterns which require or allow the strong multiplicity property

Abhilash Saha, Leona Tilis, Kevin N. Vander Meulen +1

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 th…

math.AC2013

Generalized cover ideals and the persistence property

Ashwini Bhat, Jennifer Biermann, Adam Van Tuyl

Let $I$ be a square-free monomial ideal in $R = k [x_{1}, \dots, x_{n}]$ , and consider the sets of associated primes $Ass (I^{s})$ for all integers $s \geq 1$ . Although it is known th…

Fat lines in P^3: powers versus symbolic powers

Elena Guardo, Brian Harbourne, Adam Van Tuyl

We study the symbolic and regular powers of ideals I for a family of special configurations of lines in P^3. For this family, we show that I^(m) = I^m for all integers m if and onl…

Some families of componentwise linear monomial ideals

Christopher A. Francisco, Adam Van Tuyl

Let R=k[x_1,...,x_n] be a polynomial ring over a field k. Let J={j_1,...,j_t} be a subset of [n]={1,...,n}, and let m_J denote the ideal (x_{j_1},...,x_{j_t}) of R. Given subsets J…

math.AC2013

Revisiting the spreading and covering numbers

Ben Babcock, Adam Van Tuyl

We revisit the problem of computing the spreading and covering numbers. We show a connection between some of the spreading numbers and the number of non-negative integer 2x2 matric…

math.AC2019

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

Jennifer Biermann, Christopher A. Francisco, Huy TÃ i HÃ +1

In attempting to understand how combinatorial modifications alter algebraic properties of monomial ideals, several authors have investigated the process of adding "whiskers" to gra…

Newton complementary duals of $f$ -ideals

Samuel Budd, Adam Van Tuyl

A square-free monomial ideal $I$ of $k [x_{1}, \dots, x_{n}]$ is said to be an $f$ -ideal if the facet complex and non-face complex associated with $I$ have the same $f$ -vector. We show t…

math.RA2023

Condition Numbers of Hessenberg Companion Matrices

Michael Cox, Kevin N. Vander Meulen, Adam Van Tuyl +1

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 ma…

math.CO2015

Independence complexes of well-covered circulant graphs

Jonathan Earl, Kevin N. Vander Meulen, Adam Van Tuyl

We study the independence complexes of families of well-covered circulant graphs discovered by Boros-Gurvich-MilaniÄ, Brown-Hoshino, and Moussi. Because these graphs are well-cove…

Cohen-Macaulay Circulant Graphs

Kevin N. Vander Meulen, Adam Van Tuyl, Catriona Watt

Let G be the circulant graph C_n(S) with S a subset of {1,2,...,\lfloor n/2 \rfloor}, and let I(G) denote its the edge ideal in the ring R = k[x_1,...,x_n]. We consider the problem…

math.CO2020

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

Takayuki Hibi, Kyouko Kimura, Kazunori Matsuda +1

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 poly…

Splittings of monomial ideals

Christopher A. Francisco, Huy Tai Ha, Adam Van Tuyl

We provide some new conditions under which the graded Betti numbers of a monomial ideal can be computed in terms of the graded Betti numbers of smaller ideals, thus complementing E…

math.AC2003

The Hilbert functions of ACM sets of points in P^{n_1} x ... x P^{n_k}

Adam Van Tuyl

The Hilbert functions of sets of distinct points in P^n have been characterized. We show that if we restrict to sets of distinct of points in P^{n_1} x ... x P^{n_k} that are also…

FrÃ¶berg's Theorem, vertex splittability and higher independence complexes

Priyavrat Deshpande, Amit Roy, Anurag Singh +1

A celebrated theorem of FrÃ¶berg gives a complete combinatorial classification of quadratic square-free monomial ideals with a linear resolution. A generalization of this theorem t…

math.AC2019

The regularity of some families of circulant graphs

Miguel Eduardo Uribe-Paczka, Adam Van Tuyl

We compute the Castelnuovo-Mumford regularity of the edge ideals of two families of circulant graphs, which includes all cubic circulant graphs. A feature of our approach is to com…

Simplicial complexes and Macaulay's inverse systems

Adam Van Tuyl, Fabrizio Zanello

Let $I^$ be a simplicial complex on $V = {x_{1}, ..., x_{n}}$ , with Stanley-Reisner ideal $I_{I^} \subseteq R = k [x_{1}, ..., x_{n}]$ . The goal of this paper is to investigate the class of arti…

math.CO2024

Simplicial complexes with many facets are vertex decomposable

Anton Dochtermann, Ritika Nair, Jay Schweig +2

Suppose $I^$ 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…

math.AC2003

The regularity of points in multi-projective spaces

Huy Tai Ha, Adam Van Tuyl

Let I = p_1^{m_1} \cap ... \cap p_s^{m_s} be the defining ideal of a scheme of fat points in P^{n_1} x ... x P^{n_k} with support in generic position. When all the m_i's are 1, we…

math.AC2019

Hilbert functions of schemes of double and reduced points

Enrico Carlini, Maria Virginia Catalisano, Elena Guardo +1

It remains an open problem to classify the Hilbert functions of double points in $P^{2}$ . Given a valid Hilbert function $H$ of a zero-dimensional scheme in $P^{2}$ ,…

math.CO2015

Shellability, vertex decomposability, and lexicographical products of graphs

Kevin N. Vander Meulen, Adam Van Tuyl

We investigate when the independence complex of $G [H]$ , the lexicographical product of two graphs $G$ and $H$ , is either vertex decomposable or shellable. As an application, we con…

math.AC2020

Symbolic powers of codimension two Cohen-Macaulay ideals

Susan Cooper, Giuliana Fatabbi, Elena Guardo +6

Let $I_{X}$ be the saturated homogeneous ideal defining a codimension two arithmetically Cohen-Macaulay scheme $X \subseteq P^{n}$ , and let $I_{X}^{(m)}$ denote its $m$ -th symbo…

math.AG2012

Classifying ACM sets of points in $P^{1} X P^{1}$ via separators

Elena Guardo, Adam Van Tuyl

The purpose of this note is to give a new, short proof of a classification of ACM sets of points in $P^{1} X P^{1}$ in terms of separators.

math.CO2025

Levelable graphs

Kieran Bhaskara, Michael Y. C. Chong, Takayuki Hibi +2

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 gra…

Monomial ideals, edge ideals of hypergraphs, and their graded Betti numbers

Huy Tai Ha, Adam Van Tuyl

We use the correspondence between hypergraphs and their associated edge ideals to study the minimal graded free resolution of squarefree monomial ideals. The theme of this paper is…

Partial Betti splittings with applications to binomial edge ideals

A. V. Jayanthan, Aniketh Sivakumar, Adam Van Tuyl

We introduce the notion of a partial Betti splitting of a homogeneous ideal, generalizing the notion of a Betti splitting first given by Francisco, HÃ , and Van Tuyl. Given a homog…

Distinguishing $k$ -configurations

Federico Galetto, Yong-Su Shin, Adam Van Tuyl

A $k$ -configuration is a set of points $X$ in $P^{2}$ that satisfies a number of geometric conditions. Associated to a $k$ -configuration is a sequence $(d_1…

math.AC2005

Multigraded regularity: syzygies and fat points

Jessica Sidman, Adam Van Tuyl

The Castelnuovo-Mumford regularity of a graded ring is an important invariant in computational commutative algebra, and there is increasing interest in multigraded generalizations.…

math.AC2015

On the Hilbert functions of sets of points in P^1 x P^1 x P^1

Elena Guardo, Adam Van Tuyl

Let H_X be the trigraded Hilbert function of a set X of reduced points in P^1 x P^1 x P^1. We show how to extract some geometric information about X from H_X. This note generalizes…

Powers of Principal $Q$ -Borel ideals

Eduardo Camps Moreno, Craig Kohne, Eliseo Sarmiento +1

Fix a poset $Q$ on ${x_{1}, \dots, x_{n}}$ . A $Q$ -Borel monomial ideal $I \subseteq K [x_{1}, \dots, x_{n}]$ is a monomial ideal whose monomials are closed under the Borel-like mov…

math.CO2019

Maximum nullity and zero forcing of circulant graphs

Linh Duong, Brenda K. Kroschel, Michael Riddell +2

It is well-known that the zero forcing number of a graph provides a lower bound on the minimum rank of a graph. In this paper we bound and characterize the zero forcing number of c…

math.AC2020

Regularity and h-polynomials of toric ideals of graphs

Giuseppe Favacchio, Graham Keiper, Adam Van Tuyl

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) regula…

The symbolic defect of an ideal

Federico Galetto, Anthony V. Geramita, Yong-Su Shin +1

Let $I$ be a homogeneous ideal of $k [x_{0}, \dots, x_{n}]$ . To compare $I^{(m)}$ , the $m$ -th symbolic power of $I$ , with $I^{m}$ , the regular $m$ -th power, we introduce the $m$ -th sym…

math.AC2022

Hadamard Products and Binomial Ideals

BÃ¼Åra Atar, Kieran Bhaskara, Adrian Cook +6

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 thi…

The minimal resolutions of double points in P^1 x P^1 with ACM support

Elena Guardo, Adam Van Tuyl

Let Z be a finite set of double points in P^1 x P^1 and suppose further that X, the support of Z, is arithmetically Cohen-Macaulay (ACM). We present an algorithm, which depends onl…

Associated primes of monomial ideals and odd holes in graphs

Christopher A. Francisco, Huy Tai Ha, Adam Van Tuyl

Let $G$ be a finite simple graph with edge ideal $I (G)$ . Let $J (G)$ denote the Alexander dual of $I (G)$ . We show that a description of all induced cycles of odd length in $G$ is en…

Three invariants of geometrically vertex decomposable ideals

Thai Thanh Nguyen, Jenna Rajchgot, Adam Van Tuyl

We study three invariants of geometrically vertex decomposable ideals: the Castelnuovo-Mumford regularity, the multiplicity, and the $a$ -invariant. We show that these invariants ca…

math.AC2011

Separators of Arithmetically Cohen-Macaulay fat points in P^1 x P^1

Elena Guardo, Adam Van Tuyl

Let Z be a set of fat points in P^1 x P^1 that is also arithmetically Cohen-Macaulay (ACM). We describe how to compute the degree of a separator of a fat point of multiplicity m fo…

Analytic spread of binomial edge ideals

Eduardo Camps-Moreno, Deblina Dey, Souvik Dey +5

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., clo…

Comparing invariants of toric ideals of bipartite graphs

Kieran Bhaskara, Adam Van Tuyl

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 va…

Sequentially Cohen-Macaulay bipartite graphs: vertex decomposability and regularity

Adam Van Tuyl

Let G be a bipartite graph with edge ideal I(G) whose quotient ring R/I(G) is sequentially Cohen-Macaulay. We prove: (1) the independence complex of G must be vertex decomposable,…

Balanced vertex decomposable simplicial complexes and their h-vectors

Jennifer Biermann, Adam Van Tuyl

Given any finite simplicial complex Î, we show how to construct a new simplicial complex Î_Ï that is balanced and vertex decomposable. Moreover, we show that the h-vector of the…

math.AC2008

Separators of points in a multiprojective space

Elena Guardo, Adam Van Tuyl

In this note we develop some of the properties of separators of points in a multiprojective space. In particular, we prove multigraded analogs of results of Geramita, Maroscia, and…

Regularity and $h$ -polynomials of edge ideals

Takayuki Hibi, Kazunori Matsuda, Adam Van Tuyl

For any two integers $d, r \geq 1$ , we show that there exists an edge ideal $I (G)$ such that the $reg (R / I (G))$ , the Castelnuovo-Mumford regularity of $R / I (G)$ , is $…

A classification of van der Waerden complexes with linear resolution

Takayuki Hibi, Adam Van Tuyl

In 2017, Ehrenborg, Govindaiah, Park, and Readdy defined the van der Waerden complex $vdW (n, k)$ to be the simplicial complex whose facets correspond to all the arithmetic seq…

Comparing the $v$ -number and $h$ -polynomials of edge ideals

Kamalesh Saha, Adam Van Tuyl

In this paper, we compare the $v$ -numbers and the degree of the $h$ -polynomials associated with edge ideals of connected graphs. We prove that the $v$ -number can…

math.AC2020

Homological invariants of Cameron--Walker graphs

Takayuki Hibi, Hiroju Kanno, Kyouko Kimura +2

Let $G$ be a finite simple connected graph on $[n]$ and $R = K [x_{1}, \dots, x_{n}]$ the polynomial ring in $n$ variables over a field $K$ . The edge ideal of $G$ is the ideal $I (G)$ o…

math.AG2010

Star configuration points and generic plane curves

Enrico Carlini, Adam Van Tuyl

Consider l lines in P^2 such that no three lines meet in a point. Let X(l) denote all points of intersections of these l lines. We describe all pairs (d,l) such that generic degree…

math.AC2005

Powers of complete intersections: graded Betti numbers and applications

Elena Guardo, Adam Van Tuyl

Let I = (F_1,...,F_r) be a homogeneous ideal of R = k[x_0,...,x_n] generated by a regular sequence of type (d_1,...,d_r). We give an elementary proof for an explicit description of…

Bounding invariants of fat points using a coding theory construction

Stefan O. Tohaneanu, Adam Van Tuyl

Let $Z \subseteq \proj{n}$ be a fat points scheme, and let $d (Z)$ be the minimum distance of the linear code constructed from $Z$ . We show that $d (Z)$ imposes constraints (i.e., up…

Resolutions of square-free monomial ideals via facet ideals: a survey

Huy Tai Ha, Adam Van Tuyl

We survey some recent results on the minimal graded free resolution of a square-free monomial ideal. The theme uniting these results is the point-of-view that the generators of a m…

math.AC2005

Multigraded regularity: coarsenings and resolutions

Jessica Sidman, Adam Van Tuyl, Haohao Wang

Let S = k[x_1,...,x_n] be a Z^r-graded ring with deg (x_i) = a_i \in Z^r for each i and suppose that M is a finitely generated Z^r-graded S-module. In this paper we describe how to…