Related papers: Monomial ideals and independence of $\mathrm{I}\Si…

For a joint probability density function f(x) of a random vector X the mixed partial derivatives of log f(x) can be interpreted as limiting cumulants in an infinitesimally small open neighborhood around x. Moreover, setting them to zero…

Let $S = \mathsf{k}[x_1, \ldots, x_n]$, $I$ be an ideal of $S$, and $\bar{I}$ denote its integral closure. A conjecture of K\"{u}ronya and Pintye states that for any homogeneous ideal $I$ of $S$, the inequality $\operatorname{reg}(\bar{I})…

In this paper we discuss the problem of characterizing the Cohen-Macaulay property of certain families of monomial ideals with fixed radical. More precisely, we consider generically complete intersection monomial ideals whose radical…

We will study monomial ideals $I$ in the exterior algebra as well as in the polynomial ring whose generic initial ideal is constant for all term orders up to permutations of variables. First, in the exterior algebra, we determine all graphs…

We introduce the concept of monomial ideals with stable projective dimension, as a generalization of the Cohen-Macaulay property. Indeed, we study the class of monomial ideals $I$, whose projective dimension is stable under monomial…

Let $R = k[x_1,\ldots, x_n]$ be the polynomial ring in $n$ variables over a field $k$ and let $I$ be a monomial ideal of $R$. In this paper, we study almost Cohen-Macaulay simplicial complex. Moreover, we characterize the almost…

We study the \L ojasiewicz exponent and the log canonical threshold of ideals of $\mathcal O_n$ when restricted to generic subspaces of $\mathbb C^n$ of different dimensions. We obtain effective formulas of the resulting numbers for ideals…

We study Stanley decompositions and show that Stanley's conjecture on Stanley decompositions implies his conjecture on partitionable Cohen-Macaulay simplicial complexes. We also prove these conjectures for all Cohen-Macaulay monomial ideals…

We present a procedure for computing the log-canonical threshold of an arbitrary ideal generated by binomials and monomials. The computation of the log canonical threshold is reduced to the problem of computing the minimum of a function,…

We present a class of homogeneous ideals which are generated by monomials and binomials of degree two and are set-theoretic complete intersections. This class includes certain reducible varieties of minimal degree and, in particular, the…

We consider the fiber cone of monomial ideals. It is shown that for monomial ideals $I\subset K[x,y]$ of height $2$, generated by $3$ elements, the fiber cone $F(I)$ of $I$ is a hypersurface ring, and that $F(I)$ has positive depth for…

In this paper, we introduce the concept of $k$-clean monomial ideals as an extension of clean monomial ideals and present some homological and combinatorial properties of them. Using the hierarchal structure of $k$-clean ideals, we show…

For a monomial ideal $I$, we consider the $i$th homological shift ideal of $I$, denoted by $\text{HS}_i(I)$, that is, the ideal generated by the $i$th multigraded shifts of $I$. Some algebraic properties of this ideal are studied. It is…

B. Sturmfels and S. Sullivant associated to any graph a toric ideal, called the cut ideal. We consider monomial cut ideals and we show that their algebraic properties such as the minimal primary decomposition, the property of having a…

Extending the notion of indispensable binomials of a toric ideal, we define indispensable monomials of a toric ideal and establish some of their properties. They are useful for searching indispensable binomials of a toric ideal and for…

Independent sets play a key role into the study of graphs and important problems arising in graph theory reduce to them. We define the monomial ideal of independent sets associated to a finite simple graph and describe its homological and…

Let $I \subset K[x,y]$ be a monomial ideal. How small can $\mu(I^2)$ be in terms of $\mu(I)$? It has been expected that the inequality $\mu(I^2) > \mu(I)$ should hold whenever $\mu(I) \ge 2$. Here we disprove this expectation and provide a…

In this paper we study monomial cycles in Koszul homology over a monomial ring. The main result is that a monomial cycle is a boundary precisely when the monomial representing that cycle is contained in an ideal we introduce called the…

We give a generalization of the theory of $\mathbb{Z}_2$-graded manifolds to a theory of $\mathcal{I}$-graded manifolds, where $\mathcal{I}$ is a commutative semi-ring with some additional properties. We prove Batchelor's theorem in this…

We consider ideals in the ring $\mathbb{Z}_2[x_1,\ldots, x_n]$ that contain the polynomials $x_i^2 - x_i$ for $i = 1, \ldots, n$ and give various results related to the one-to-one correspondence between these ideals and the subsets of…