Related papers: $p-$Ferrer diagram, $p-$linear ideals and arithmet…

Utilizing elliptic weights, we construct an elliptic analogue of rook numbers for Ferrers boards. Our elliptic rook numbers generalize Garsia and Remmel's q-rook numbers by two additional independent parameters a and b, and a nome p. These…

We give a lower bound for the Castelnuovo-Mumford regularity of binomial edge ideals of block graphs by computing the two distinguished extremal Betti numbers of a new family of block graphs, called flower graphs. Moreover, we present a…

This paper exhibits some new examples of the behavior of the Castelnuovo-Mumford regularity of homogeneous ideals in polynomial rings. More precisely, we present new examples of homogenous ideals with large regularity compared to the…

We classify generalized block graphs whose binomial edge ideals admit a unique extremal Betti number. We prove that the Castelnuovo-Mumford regularity of binomial edge ideals of generalized block graphs is bounded below by $m(G)+1$, where…

Let $S={\sf k}[X_1,\dots, X_n]$ be a polynomial ring, where ${\sf k}$ is a field. This article deals with the defining ideal of the Rees algebra of squarefree monomial ideal generated in degree $n-2$. As a consequence, we prove that Betti…

The $\mathrm{v}$-number of a graded ideal $I\subseteq R$, denoted by $\mathrm{v}(I)$, is the minimum degree of a polynomial $f$ for which $I:f$ is a prime ideal. Jaramillo and Villarreal (J Combin Theory Ser A 177:105310, 2021) studied the…

In this paper, we introduced the concept of a $p$-ideal for a given ring. We provide necessary and sufficient condition for $\dfrac{R[x]}{(f(x))}$ to be a $p$-ring, where $R$ is a finite $p$-ring. It is also shown that the amalgamation of…

Let $K$ be an algebraically closed field of null characteristic and $p(z)$ a Hilbert polynomial. We look for the minimal Castelnuovo-Mumford regularity $m_{p(z)}$ of closed subschemes of projective spaces over $K$ with Hilbert polynomial…

In this paper, we study the algebra of Veronese type. We show that the presentation ideal of this algebra has an initial ideal whose Alexander dual has linear quotients. As an application, we explicitly obtain the Castelnuovo-Mumford…

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…

We study the binomial and monomial ideals arising from linear equivalence of divisors on graphs from the point of view of Gr\"obner theory. We give an explicit description of a minimal Gr\"obner bases for each higher syzygy module. In each…

Let $p$ be an odd prime number. In this paper, we construct $2(2p-3)$ classes of codes over the ring $R=\Bbb F_p+u\Bbb F_p,u^2=0$, which are associated with down sets. We compute the Lee weight distributions of the $2(2p-3)$ classes of…

In a 2008 paper, the first author and Van Tuyl proved that the regularity of the edge ideal of a graph G is at most one greater than the matching number of G. In this note, we provide a generalization of this result to any square-free…

We study when blowup algebras are $F$-split or strongly $F$-regular. Our main focus is on algebras given by symbolic and ordinary powers of ideals of minors of a generic matrix, a symmetric matrix, and a Hankel matrix. We also study ideals…

Let $(R, \mathfrak m)$ be a Cohen-Macaulay local ring of dimension $d \geq 2,$ and $I$ an $\mathfrak m$-primary ideal of $R.$ Denote $r_{J}(I)$ as the reduction number of $I$ with respect to a minimal reduction $J$ of $I,$ and $\rho(I)$ as…

We study the vanishing ideal of the parametrized algebraic toric associated to the complete multipartite graph $\G=\mathcal{K}_{\alpha_1,...,\alpha_r}$ over a finite field of order $q$. We give an explicit family of binomial generators for…

The aim of this paper is to make a connection between design theory and algebraic geometry/commutative algebra. In particular, given any Steiner System $S(t,n,v)$ we associate two ideals, in a suitable polynomial ring, defining a Steiner…

Let $C$ be a closed cone with nonempty interior $C^\circ$ in a Banach space. Let $f:C^\circ \rightarrow C^\circ$ be an order-preserving subhomogeneous function with a fixed point in $C^\circ$. We introduce a condition which guarantees that…

We establish strong relationships between the Castelnuovo-Mumford regularity and the Ratliff-Rush closure of an ideal. Our results have several interesting consequences on the computation of the Ratliff-Rush closure, the stability of the…

Two boards are rook equivalent if they have the same number of non-attacking rook placements for any number of rooks. Define a rook equivalence graph of an equivalence set of Ferrers boards by specifying that two boards are connected by an…