Related papers: First Nonlinear Syzygies of Ideals Associated to G…

The index of a graded ideal measures the number of linear steps in the graded minimal free resolution of the ideal. In this paper we study the index of powers and squarefree powers of edge ideals. Our results indicate that the index as a…

We prove a duality theorem for simplicial complexes arising from a combinatorial construction we define, which applies to the squarefree monomial complexes for Veronese ideals of projective spaces and weighted projective spaces. Our theorem…

We study the syzygies of secant ideals of Veronese subrings of a fixed commutative graded algebra over a field of characteristic 0. One corollary is that the degrees of the minimal generators of the ith syzygy module of the coordinate ring…

We study algebraic and homological properties of facet ideals of order complexes of posets which we call ideals of maximal flags of posets or simply flag ideals. We characterize the unmixed and Cohen-Macaulay flag ideals of graded posets.…

Let G be a numerical semigroup. In this paper, we prove an upper bound for the Betti numbers of the semigroup ring of G which depends only on the width of G, that is, the difference between the largest and the smallest generator of G. In…

Let S be a polynomial ring and R=S/I where I is a graded ideal of S. The Multiplicity Conjecture of Herzog, Huneke, and Srinivasan which was recently proved using the Boij-Soederberg theory states that the multiplicity of R is bounded above…

The ideals generated by fold products of linear forms are generalizations of powers of defining ideals of star configurations, or of Veronese type ideals, and in this paper we study their Betti numbers. In earlier work, the authors together…

A new approach is established to computing the image of a rational map, whereby the use of approximation complexes is complemented with a detailed analysis of the torsion of the symmetric algebra in certain degrees. In the case the map is…

Let $A$ and $B$ be standard graded polynomial rings over a field $k$ and $I$ and $J$ be non-zero, proper homogeneous ideals contained in $A$ and $B$, respectively. Denote by $P$ the sum of $I$ and $J$ in $R=A\otimes_k B$. Under reasonable…

In this paper we give new upper bounds on the regularity of edge ideals whose resolutions are k-steps linear; surprisingly, the bounds are logarithmic in the number of variables. We also give various bounds for the projective dimension of…

The singularity degree of a semidefinite programming problem is the smallest number of facial reduction steps to make the problem strictly feasible. We introduce two new graph parameters, called the singularity degree and the nondegenerate…

Let K be a field and let S = K[x_1, ..., x_n] be a polynomial ring. Consider a homogenous ideal I in S. Let t_i denote reg(Tor_i (S/I, K)), the maximal degree of an ith syzygy of S/I. We prove bounds on the numbers t_i for i > n/2 purely in…

Given integers $r$ and $b$ with $1 \leq b \leq r$, a finite simple connected graph $G$ for which ${\rm reg}(S/I(G)) = r$ and the number of extremal Betti numbers of $S/I(G)$ is equal to $b$ will be constructed.

Based on Bergman's Lemma on centralizers, we obtain a sharp lower degree bound for nonconstant elements in a subalgebra generated by two elements of a free associative algebra over an arbitrary field.

In this paper we investigate the class of rigid monomial ideals. We give a characterization of the minimal free resolutions of certain classes of these ideals. Specifically, we show that the ideals in a particular subclass of rigid monomial…

The conjecture of Kalai, Kleinschmidt, and Lee on the number of empty simplices of a simplicial polytope is established by relating it to the first graded Betti numbers of the polytope. The proof allows us to derive explicit optimal bounds…

We consider Stanley--Reisner rings $k[x_1,...,x_n]/I(\mc{H})$ where $I(\mc{H})$ is the edge ideal associated to some particular classes of hypergraphs. For instance, we consider hypergraphs that are natural generalizations of graphs that…

Suppose that G is a linearly reductive group. We study the minimal free resolution of the invariant ring. If G is a finite linearly reductive group, then the ring of invariants is generated in degree at most |G|, the group order. We prove…

We introduce a novel approach named the {\it induced-subgraph approach} for investigating the Betti numbers of normal edge rings. Utilizing this approach, we compute all the multi-graded Betti numbers of the edge rings associated with…

We study monomial cut ideals associated to a graph $G$, which are a monomial analogue of toric cut ideals as introduced by Sturmfels and Sullivant. Primary decompositions, projective dimensions, and Castelnuovo-Mumford regularities are…