Related papers: A class of square-free monomial ideals associated …

Fix a square-free monomial $m \in S = \mathbb{K}[x_1,\ldots,x_n]$. The square-free principal Borel ideal generated by $m$, denoted ${\rm sfBorel}(m)$, is the ideal generated by all the square-free monomials that can be obtained via Borel…

We consider rings whose one-sided ideals are close to automorphism-invariant modules. We study rings in which every (finitely generated) right ideal is automorphism invariant and rings in which every right ideal is a finite direct sum of…

For an ideal $I_{m,n}$ generated by all square-free monomials of degree $m$ in a polynomial ring $R$ with $n$ variables, we obtain a specific embedding of a canonical module of $R/I_{m,n}$ to $R/I_{m,n}$ itself. The construction of this…

We survey a number of recent studies of the Castelnuovo-Mumford regularity of squarefree monomial ideals. Our focus is on bounds and exact values for the regularity in terms of combinatorial data from associated simplicial complexes and/or…

Let $I$ be a square-free monomial ideal in a polynomial ring $R=K[x_1,\ldots, x_n]$ over a field $K$, $\mathfrak{m}=(x_1, \ldots, x_n)$ be the graded maximal ideal of $R$, and $\{u_1, \ldots, u_{\beta_1(I)}\}$ be a maximal independent set…

For an algebraic number $\alpha$ of degree $n$, let $\mathcal{M}_{\alpha}$ be the $\mathbb{Z}$-module generated by $1,\alpha ,\ldots ,\alpha^{n-1}$; then $\mathbb{Z}_{\alpha}:=\{\xi\in\mathbb{Q} (\alpha ):\,…

Let $I\supsetneq J$ be two square free monomial ideals of a polynomial algebra over a field generated in degree $\geq 1$, resp. $\geq 2$ . Almost always when $I$ contains precisely one variable, the other generators having degrees $\geq 2$,…

In [7] we obtained a formula for the Hilbert depth of squarefree Veronese ideals in a standard graded polynomial ring by relating it to the Hilbert depth of powers of the irrelevant maximal ideal. In this paper, we prove that these two…

Consider the ring $\mathcal{S}$ of symmetric polynomials in $k$ variables over an arbitrary base ring $\mathbf{k}$. Fix $k$ scalars $a_{1},a_{2},\ldots,a_{k}\in\mathbf{k}$. Let $I$ be the ideal of $\mathcal{S}$ generated by…

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…

We define a set of invariants of a homogeneous ideal $I$ in a polynomial ring called the symmetric iterated Betti numbers of $I$. For $I_{\Gamma}$, the Stanley-Reisner ideal of a simplicial complex $\Gamma$, these numbers are the symmetric…

We study the licci property for several classes of squarefree monomial ideals arising from graphs and related combinatorial structures. We characterize licci bi-Cohen-Macaulay squarefree monomial ideals, complementary edge ideals, $t$-path…

The free resolution and the Alexander dual of squarefree monomial ideals associated with certain subsets of distributive lattices are studied.

A \textit{symmetric ideal} $I \subseteq R = K[x_1,x_2,...]$ is an ideal that is invariant under the natural action of the infinite symmetric group. We give an explicit algorithm to find Gr\"obner bases for symmetric ideals in the infinite…

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…

We continue the study of intersection algebras $\mathcal B = \mathcal B_R(I, J)$ of two ideals $I, J$ in a commutative Noetherian ring $R$. In particular, we exploit the semigroup ring and toric structures in order to calculate various…

We use initially regular sequences that consist of linear sums to explore the depth of $R/I^2$, when $I$ is a monomial ideal in a polynomial ring $R$. We give conditions under which these linear sums form regular or initially regular…

Let $R=\mathbf{C}[\xi_1,\xi_2,\ldots]$ be the infinite variable polynomial ring, equipped with the natural action of the infinite symmetric group $\mathfrak{S}$. We classify the $\mathfrak{S}$-primes of $R$, determine the containments among…

We solve a class of isoperimetric problems on $\mathbb{R}^2_+ :=\left\{ (x,y)\in \mathbb{R} ^2 : y>0 \right\}$ with respect to monomial weights. Let $\alpha $ and $\beta $ be real numbers such that $0\le \alpha <\beta+1$, $\beta\le 2…

We compute the integral of monomials of the form $x^{2\beta}$ over the unit sphere and the unit ball in $R^n$ where $\beta = (\beta_1,...,\beta_n)$ is a multi-index with real components $\beta_k > -1/2$, $1 \le k \le n$, and discuss their…