Related papers: Homogeneous Liaison and the Sequentially Bounded L…
Following G.~Gr\"atzer and E.~Knapp, 2009, a planar semimodular lattice $L$ is \emph{rectangular}, if~the left boundary chain has exactly one doubly-irreducible element, $c_l$, and the right boundary chain has exactly one doubly-irreducible…
Let $G$ be a simple graph on $n$ vertices and $\mathcal{I}_G$ denotes parity binomial edge ideal of $G$ in the polynomial ring $S = \mathbb{K}[x_1,\ldots, x_n, y_1, \ldots, y_n].$ We obtain a lower bound for the regularity of parity…
Given a $G$-invariant potential $\mathcal{V}$ of a scalar multiplet $\varphi$, there may exist a set of homogenous linear equations that constrain the components of a stationary point of $\mathcal{V}$ independently of the coefficients of…
Let $A$ be a unitary ring and let $(\mathbf{I(A),\subseteq })$ be the lattice of ideals of the ring $A.$ In this article we will study the property of the lattice $(\mathbf{I(A),\subseteq})$ to be Noetherian or not, for various types of…
We study the complete intersection property and the algebraic invariants (index of regularity, degree) of vanishing ideals on degenerate tori over finite fields. We establish a correspondence between vanishing ideals and toric ideals…
To a natural number $n$, a finite partially ordered set $P$ and a poset ideal ${\mathcal J}$ in the poset $Hom(P,[n])$ of isotonian maps from $P$ to the chain on $n$ elements, we associate two monomial ideals, the letterplace ideal…
We consider the "limiting behavior" of *discriminants*, by which we mean informally the locus in some parameter space of some type of object where the objects have certain singularities. We focus on the space of partially labeled points on…
Let I be an m-primary ideal of a Noetherian local ring (R,m). We consider the Gorenstein and complete intersection properties of the associated graded ring G(I) and the fiber cone F(I) of I as reflected in their defining ideals as…
An ideal $I$ of a commutative ring $R$ is said to be of linear type when its Rees algebra and symmetric algebra exhibit isomorphism. In this paper, we investigate the conjecture put forth by Jayanthan, Kumar, and Sarkar (2021) that if $G$…
We investigate, using the notion of linear quotients, significative classes of connected graphs whose monomial edge ideals, not necessarily squarefree, have linear resolution, in order to compute standard algebraic invariants of the…
We study basic properties of monomial ideals with linear quotients. It is shown that if the monomial ideal $I$ has linear quotients, then the squarefree part of $I$ and each component of $I$ as well as $\mm I$ have linear quotients, where…
In combinatorial commutative algebra and algebraic statistics many toric ideals are constructed from graphs. Keeping the categorical structure of graphs in mind we give previous results a more functorial context and generalize them by…
We show that the regularity of monomial ideals whose associated prime ideals are totally ordered by inclusion is linearly bounded.
The Hilbert polynomial of a homogeneous complete intersection is determined by the degrees of the generators of the defining ideal. The degrees of the generators are not, in general, determined by the Hilbert polynomial -- but sometimes…
Hetyei introduced in 2019 the homogenized Linial arrangement and showed that its regions are counted by the median Genocchi numbers. In the course of devising a different proof of Hetyei's result, Lazar and Wachs considered another…
The method of imsets, introduced by Studen\'y, provides a geometric and combinatorial description of conditional independence statements. Elementary conditional independence statements over a finite set of discrete random variables…
In this paper we prove parts of a conjecture of Herzog giving lower bounds on the rank of the free modules appearing in the linear strand of a graded $k$-th syzygy module over the polynomial ring. If in addition the module is…
The Boolean ring $B$ of measurable subsets of the unit interval, modulo sets of measure zero, has proper radical ideals (e.g., $\{0\})$ that are closed under the natural metric, but has no prime ideals closed under that metric; hence closed…
We study monomial ideals with linear presentation or partially linear resolution. We give combinatorial characterizations of linear presentation for square-free ideals of degree 3, and for primary ideals whose resolutions are linear except…
We present two new problems on lower bounds for resolution Betti numbers of monomial ideals generated in a fixed degree. The first concerns any such ideal and bounds the total Betti numbers, while the second concerns ideals that are…