Related papers: Behaviour of the normalized depth function

For every graph $G$, let $\alpha(G)$ denote its independence number. What is the minimum of the maximum degree of an induced subgraph of $G$ with $\alpha(G)+1$ vertices? We study this question for the $n$-dimensional Hamming graph over an…

Every monomial ideal $I$ has a Scarf complex, which is a subcomplex of its minimal free resolution. We say that $I$ is Scarf if its Scarf complex is also its minimal free resolution. In this paper, we fully characterize all pairs $(G,n)$ of…

Let $G$ be a graph on the vertex set $[n]$ and $J_G$ the associated binomial edge ideal in the polynomial ring $S=\mathbb{K}[x_1,\ldots,x_n,y_1,\ldots,y_n]$. In this paper we investigate the depth of binomial edge ideals. More precisely, we…

Let $I$ be a monomial ideal in a polynomial ring $S=K[x_1,\ldots,x_n]$ over a field $K$ with $n=2$ or $3$, and let $\overline{I}$ be its integral closure. We will show that $\text{reg} (\overline{I}) \le \text{reg} (I)$. Furthermore, if $I$…

A monomial ideal $I$ is said to have homological linear quotients if for each $k\geq 0$, the homological shift ideal $\mathrm{HS}_k(I)$ has linear quotients. It is a well-known fact that if an edge ideal $I(G)$ has homological linear…

A monomial ideal $I\subseteq \mathbb{K}[x_1,\ldots , x_n]$ is called a Simis ideal if $I^{(s)}=I^s$ for all $s\geq 1$, where $I^{(s)}$ denotes the $s$-th symbolic power of $I$. Let $I$ be a support-2 monomial ideal such that its irreducible…

Let $I$ be an ideal of a polynomial algebra $S$ over a field generated by square free monomials of degree $\geq d$. If $I$ contains more monomials of degree $d$ than $(n-d)/(n-d+1)$ of the total number of square free monomials of $S$ of…

A square-free monomial ideal $I$ is called an {\it $f$-ideal}, if both $\delta_{\mathcal{F}}(I)$ and $\delta_{\mathcal{N}}(I)$ have the same $f$-vector, where $\delta_{\mathcal{F}}(I)$ ($\delta_{\mathcal{N}}(I)$, respectively) is the facet…

Let $J\subsetneq I$ be two ideals of a polynomial ring $S$ over a field, generated by square free monomials. We show that some inequalities among the numbers of square free monomials of $I\setminus J$ of different degrees give upper bounds…

Given a $d \times n$ integer matrix $A$, the main result is an elementary, simple-to-state algorithm that finds the largest $A$-graded ideal contained in any ideal $I$ in a polynomial ring $\Bbbk[x_1,\ldots,x_n]$. The special case where $A$…

Let $G$ be a $k$-degenerate graph of order $n.$ It is well-known that $G\ $has no more edges than $S_{n,k},$ the join of a complete graph of order $k$ and an independent set of order $n-k.$ In this note it is shown that $S_{n,k}$ is…

We investigate symbolic and regular powers of monomial ideals. For a square-free monomial ideal $I$ in $k[x_0, \ldots, x_n]$ we show $I^{t(m+e-1)-e+r)}$ is a subset of $M^{(t-1)(e-1)+r-1}(I^{(m)})^t$ for all positive integers $m$, $t$ and…

For a finite simple graph $G$ and an integer $r \ge 1$, the $r$-connected ideal $I_r(G)$ is the squarefree monomial ideal generated by the vertex sets of connected induced subgraphs of size $r+1$, extending the classical edge ideal. We…

Let $G$ be a simple graph of order $n$. An independent set in a graph is a set of pairwise non-adjacent vertices. The independence polynomial of $G$ is the polynomial $I(G,x)=\sum_{k=0}^{n} s(G,k) x^{k}$, where $s(G,k)$ is the number of…

Let $R=\oplus_{\Gamma\in\Gamma}R_{\gamma}$ be a $\Gamma$-graded $K$-algebra over a field $K$, where $\Gamma$ is a totally ordered semigroup, and let $I$ be an ideal of $R$. Considering the $\Gamma$-grading filtration $FR$ of $R$ and the…

There is a natural infinite graph whose vertices are the monomial ideals in a polynomial ring. The definition involves Gr\"obner bases or the action of an algebraic torus. We present algorithms for computing the (affine schemes…

The degree excess function $\epsilon(I;n)$ is the difference between the maximal generating degree $d(I^n)$ of a homogeneous ideal $I$ of a polynomial ring and $p(I)n$, where $p(I)$ is the leading coefficient of the asymptotically linear…

If alpha=alpha(G) is the maximum size of an independent set and s_{k} equals the number of stable sets of cardinality k in graph G, then I(G;x)=s_{0}+s_{1}x+...+s_{alpha}x^{alpha} is the independence polynomial of G. In this paper we prove…

Let $\mathbb{K}$ be a field and $S=\mathbb{K}[x_1,...,x_n]$ be the polynomial ring in $n$ variables over the field $\mathbb{K}$. For every monomial ideal $I\subset S$, We provide a recursive formula to determine a lower bound for the…

Given a homogeneous ideal I of a polynomial ring A=K[X_1,...,X_n] and a monomial order, we construct a new monomial ideal of A associated with I. We call it the zero-generic initial ideal of I with respect to the order and denote it with…