Related papers: The saturation number of monomial ideals

The support poset of a monomial ideal $I\subseteq\mathbf{k}[x_1,\dots,x_n]$ encodes the relation between the variables $x_1,\dots,x_n$ and the minimal monomial generators of $I$. It is known that not every poset is realizable as the support…

Let R=k[x_1,...,x_n] be a polynomial ring over a field k. Let J={j_1,...,j_t} be a subset of [n]={1,...,n}, and let m_J denote the ideal (x_{j_1},...,x_{j_t}) of R. Given subsets J_1,...,J_s of [n] and positive integers a_1,...,a_s, we…

Let $S=K[x_1, \ldots,x_n]$ denote the polynomial ring in $n$ variables over a field $K$ and $I \subset S$ a monomial ideal. Given a vector $\mathfrak{c}\in\mathbb{N}^n$, the ideal $I_{\mathfrak{c}}$ is the ideal generated by those monomials…

Let $S$ be the power series ring or the polynomial ring over a field $K$ in the variables $x_1,\ldots,x_n$, and let $R=S/I$, where $I$ is proper ideal which we assume to be graded if $S$ is the polynomial ring. We give an explicit…

After sketching the basic theory of injective ideals of homogeneous polynomials, we characterize injective polynomial ideals by means of a domination property and applications of this characterization to some classical operator ideals and…

In this paper we consider symmetric powers representation and exterior powers representation of finite groups, which generated by the representation which has finite dimension over the complex field. We calculate the multiplicity of…

The main focus of this paper is on the problem of relating an ideal $I$ in the polynomial ring $\mathbb Q[x_1, \dots, x_n]$ to a corresponding ideal in $\mathbb F_p[x_1,\dots, x_n]$ where $p$ is a prime number; in other words, the…

Let $S = k[x_{11}, \cdots, x_{1b_1}, \cdots, x_{n1}, \cdots, x_{nb_n}]$ be a polynomial ring in $m = b_1 + \cdots + b_n$ variables over a field $k$. For all $j$, $1\le j \le n$, let $P_j$ be the prime ideal generated by variables $\{x_{j1},…

Let $I\subset A$ and $J\subset B$ be two monomial ideals, where $A$ and $B$ are two polynomial rings with disjoint variables. Considering a general set-up of monomial filtrations, we study the behaviour of the $\mathrm{v}$-function under…

Let $R$ and $S$ be standard graded algebras over a field $k$, and $I \subseteq R$ and $J \subseteq S$ homogeneous ideals. Denote by $P$ the sum of the extensions of $I$ and $J$ to $R\otimes_k S$. We investigate several important homological…

We prove that a monomial ideal $I$ generated in a single degree, is polymatroidal if and only if it has linear quotients with respect to the lexicographical ordering of the minimal generators induced by every ordering of variables. We also…

Let $B$ be a reduced local (Noetherian) ring with maximal ideal $M$. Suppose that $B$ contains the rationals, $B/M$ is uncountable and $|B| = |B/M|$. Let the minimal prime ideals of $B$ be partitioned into $m \geq 1$ subcollections $C_1,…

In this article we study inequalities of ideal norms. We prove that in a subring $R$ of a number field every ideal can be generated by at most $3$ elements if and only if the ideal norm satisfies $N(IJ) \geq N(I)N(J)$ for every pair of…

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…

Let $I$ be a two-dimensional squarefree monomial ideal of a polynomial ring $S$. We evaluate the geometric regularity, $a_i$-invariants for $i\geq 1$ of the power $I^n$. It turns out they are all linear functions in $n$ from $n=2$.…

For any ideal $I$ in a Noetherian local ring or any graded ideal $I$ in a standard graded $K$-algebra over a field $K$, we introduce the socle module $\mathrm{Soc}(I)$, whose graded components give us the socle of the powers of $I$. It is…

The paper presents two algorithms for finding irreducible decomposition of monomial ideals. The first one is recursive, derived from staircase structures of monomial ideals. This algorithm has a good performance for highly non-generic…

In this paper, we are concerned with the problem of counting the multiplicities of a zero-dimensional regular set's zeros. We generalize the squarefree decomposition of univariate polynomials to the so-called pseudo squarefree decomposition…

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…

Vanishing polynomials are polynomials over a ring which output $0$ for all elements in the ring. In this paper, we study the ideal of vanishing polynomials over specific types of rings, along with the closely related ring of polynomial…