Related papers: t-spread strongly stable monomial ideals

Let $K$ be a field and $S=K[x_1,\ldots,x_n]$ a standard polynomial ring over $K$. In this paper, some new optimized algorithms to compute the smallest $t$-spread lexicographic set and the smallest $t$-spread strongly stable set containing a…

We introduce the class of vector-spread monomial ideals. This notion generalizes that of $t$-spread ideals introduced by Ene, Herzog and Qureshi. In particular, we focus on vector-spread strongly stable ideals, we compute their Koszul…

Let $K$ be a field and let $S=K[x_1,\dots,x_n]$ be a standard polynomial ring over a field $K$. We characterize the extremal Betti numbers, values as well positions, of a $t$-spread strongly stable ideal of $S$. Our approach is…

We introduce and study monomial ideals with regular quotients, which can be seen as an extension of monomial ideals with linear quotients. Based on these investigations, we are able to calculate the Betti numbers of toric ideals belonging…

Strongly stable ideals are a class of monomial ideals which correspond to generic initial ideals in characteristic zero and can be described completely by their Borel generators, a subset of the minimal monomial generators of the ideal.…

In this paper, our main focus is to explore different classes of nearly normally torsion-free ideals. We first characterize all finite simple connected graphs with nearly normally torsion-free cover ideals. Next, we characterize all…

Let $S=K[x_1,\dots,x_n]$ be a polynomial ring in $n$ variables with coefficients over a field $K$. A $t$-spread lexsegment ideal $I$ of $S$ is a monomial ideal generated by a $t$-spread lexsegment set. We determine all $t$-spread lexsegment…

We prove that $t$-spread principal Borel ideals are sequentially Cohen-Macaulay and study their powers. We show that these ideals possess the strong persistence property and compute their limit depth.

We study the extremal Betti numbers of the class of $t$--spread strongly stable ideals. More precisely, we determine the maximal number of admissible extremal Betti numbers for such ideals, and thereby we generalize the known results for…

Let $S=K[x_1,\dots,x_n]$ be the standard graded polynomial ring, with $K$ a field, and let ${\bf t}=(t_1,\ldots,t_{d-1})\in{\mathbb{Z}}_{\ge 0}^{d-1}$, $d\ge 2$, be a $(d-1)$-tuple whose entries are non negative integers. To a ${\bf…

We introduce the notion of sortability and $t$-sortability for a simplicial complex and study the graphs for which their independence complexes are either sortable or $t$-sortable. We show that the proper interval graphs are precisely the…

We prove that any t-spread strongly stable ideal has a unique t-spread lex ideal with the same f-vector. We also characterize the possible f-vectors of t-spread strongly stable ideals in the "t-spread" analogue of Kruskal-Katona theorem.

Strongly stable monomial ideals are important in algebraic geometry, commutative algebra, and combinatorics. Prompted, for example, by combinatorial approaches for studying Hilbert schemes and the existence of maximal total Betti numbers…

An equigenerated monomial ideal $I$ is a Freiman ideal if $\mu(I^2)=\ell(I)\mu(I)-{\ell(I)\choose 2}$ where $\ell(I)$ is the analytic spread of $I$ and $\mu(I)$ is the least number of monomial generators of $I$. Freiman ideals are special…

When a monomial ideal has linear quotients with respect to an admissible order of increasing support-degree, we provide two proofs of different flavors to show that it is componentwise support-linear. We also introduce the variable…

We study ideals which are generated by monomials of degree $d$ in the polynomial ring in $n$ variables and which satisfy certain numerical side conditions regarding their exponents. Typical examples of such ideals are the ideals of Veronese…

It is shown that a squarefree principal Borel ideal satisfies the persistence property for the associated prime ideals. For the graded maximal ideal we compute the index of stability with respect to squarefree principal Borel ideals and…

Let $K$ be a field and $S = K[x_1,\dots,x_n]$ be a polynomial ring over $K$. We discuss the behaviour of the extremal Betti numbers of the class of squarefree strongly stable ideals. More precisely, we give a numerical characterization of…

In this paper, by a modification of a previously constructed minimal free resolution for a transversal monomial ideal, the Betti numbers of this ideal is explicitly computed. For convenient characteristics of the ground field, up to a…

We consider vector-spread Borel ideals. We show that these ideals have linear quotients and thereby we determine the graded Betti numbers and the bigraded Poincar\'e series. A characterization of the extremal Betti numbers of such a class…