相关论文: Realizing resolutions of powers of extremal ideals

We study algebraic and homological properties of the ideal of submaximal minors of a sparse generic symmetric matrix. This ideal is generated by all $(n-1)$-minors of a symmetric $n \times n$ matrix whose entries in the upper triangle are…

Let $I(G)$ be the edge ideal of a gapfree graph $G$. An open conjecture of Nevo and Peeva states that $I(G)^q$ has linear resolution for $q\gg 0$. We present a promising approach to this challenging conjecture by investigating the stronger…

We discuss extremal Betti tables of resolutions in three different contexts. We begin over the graded polynomial ring, where extremal Betti tables correspond to pure resolutions. We then contrast this behavior with that of extremal Betti…

Motivated by using combinatorics to study jets of monomial ideals, we extend a definition of jets from graphs to clutters. We offer some structural results on their vertex covers, and show an interesting connection between the cover ideal…

It is known that Paley graphs of square order have the strict-EKR property, that is, all maximum cliques are canonical cliques. Peisert-type graphs are natural generalizations of Paley graphs and some of them also have the strict-EKR…

The aim of this work is to compare symbolic and ordinary powers of monomial ideals using commutative algebra and combinatorics. Monomial ideals whose symbolic and ordinary powers coincide are called Simis ideals. Weighted monomial ideals…

Let $I\subset R=K[x_1, \ldots, x_n]$ be a square-free monomial ideal, $\mathfrak{q}$ be a prime monomial ideal in $R$, $h$ be a square-free monomial in $R$ with $\mathrm{supp}(h) \cap (\mathrm{supp}(\mathfrak{q}) \cup…

We investigate ideals in a polynomial ring which are generated by powers of linear forms. Such ideals are closely related to the theories of fat point ideals, Cox rings, and box splines. We pay special attention to a family of power ideals…

We develop an algebraic theory of supports for $R$-linear codes of fixed length, where $R$ is a finite commutative unitary ring. A support naturally induces a notion of generalized weights and allows one to associate a monomial ideal to a…

We study the regularity of symbolic powers of square-free monomial ideals. We prove that if $I = I_\Delta$ is the Stanley-Reisner ideal of a simplicial complex $\Delta$, then $\reg(I^{(n)}) \leqslant \delta(n-1) +b$ for all $n\geqslant 1$,…

Let $G$ be a finite simple graph and let $I(G)$ be its edge ideal. In this article, we deeply investigate the squarefree powers of $I(G)$ by means of Betti splittings. When $G$ is a forest, it is shown that the normalized depth function of…

Let $G$ be a cycle or a complete bipartite graph. We show that the binomial edge ideal $J_{G}$ and its initial ideal with respect to the lexicographic order have the same extremal Betti number.

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 Eliahou and Kervaire's splitting approach. As applications,…

Let $S$ be a polynomial ring in $n$ variables over a field $K$ of characteristic $0$. A numerical characterization of all possible extremal Betti numbers of any graded submodule of a finitely generated graded free $S$-module is given.

We introduce a general random model of a combinatorial optimization problem with geometric structure that encapsulates both linear programming and integer linear programming. Let $Q$ be a bounded set called the feasible set, $E$ be an…

Let I be a finitely supported complete m-primary ideal of a regular local ring (R, m). A theorem of Lipman implies that I has a unique factorization as a *-product of special *-simple complete ideals with possibly negative exponents for…

Zonotopal algebra deals with ideals and vector spaces of polynomials that are related to several combinatorial and geometric structures defined by a finite sequence of vectors. Given such a sequence X, an integer k>=-1 and an upper set in…

An ideal of polynomials is symmetric if it is closed under permutations of variables. We relate general symmetric ideals to the so called Specht ideals generated by all Specht polynomials of a given shape. We show a connection between the…

In this article, we investigate the $\operatorname{v}$-numbers of powers of monomial ideals and their integral closures in a polynomial ring $S$. We provide an alternative proof for determining the $\operatorname{v}$-numbers of powers of…

The Minimal Resolution Conjecture (MRC) for points on a projective variety X predicts that the Betti numbers of general sets of points in X are as small as the geometry (Hilbert function) of X allows. To a large extent, we settle this…