Related papers: Monomial ideals with tiny squares

For a simplicial complex X and a field K, let h_i(X)=\dim \tilde{H}_i(X;K). It is shown that if X,Y are complexes on the same vertex set, then for all k h_{k-1}(X\cap Y) \leq \sum_{\sigma \in Y} \sum_{i+j=k} h_{i-1}(X[\sigma])\cdot…

We consider a (possibly discrete) unimodular locally compact group $G$ with Haar measure $\mu_G$, and a compact $A\subseteq G$ of positive measure with $\mu_G(A^2)\leq K\mu_G(A)$. Let $H$ be a closed normal subgroup of G and $\pi: G…

We construct monomial ideals with the property that their depth function has any given number of strict local maxima.

Let $(R,\mathfrak{m})$ be a regular local ring or a polynomial ring over a field, and let $I$ be an ideal of $R$ which we assume to be graded if $R$ is a polynomial ring. Let astab$(I)$ resp. $\overline{\rm astab}(I)$ be the smallest…

Given two toric ideals $I_1,I_2\subset\si$, it is not always true that $I_1+I_2$ is a toric ideal. Given $I_1,...,I_k\subset\si$ a familly of toric ideals we give necessary conditions in order to have that $I_1+...+I_k$ is a toric ideal.

Let $S=K[x_1,\ldots,x_n]$ be the ring of polynomials over a field $K$. Given two monomial ideals $0\subset I\subsetneq J \subset S$, we present a new method to compute the Hilbert depth of $J/I$. As an application, we show that if $u\in S$…

We show that Stanley's conjecture holds for a polynomial ring over a field in four variables. In the case of polynomial ring in five variables, we prove that the monomial ideals with all associated primes of height two, are Stanley ideals.

Classical result states that for two continuous and strict means $M,\,N \colon I^2 \to I$ ($I$ is an interval) there exists a unique $(M,N)$-invariant mean $K \colon I^2 \to I$, i.e. such a mean that $K \circ (M,N)=K$ and, moreover, the…

There is a one-to-one correspondence between square-free monomial ideals and clutters, which are also known as simple hypergraphs. It was conjectured that unmixed admissible clutters are Cohen-Macaulay. We prove the conjecture for uniform…

We establish a lower bound for the frequency with which an irreducible monic cubic polynomial with negative discriminant can be expressed as a sum of two squares ($\square_{2}$). This provides a quantitative answer to a question posed by…

Let K be a field and let S = K[x_1, ..., x_n] be a polynomial ring. Consider a homogenous ideal I in S. Let t_i denote reg(Tor_i (S/I, K)), the maximal degree of an ith syzygy of S/I. We prove bounds on the numbers t_i for i > n/2 purely in…

For compact Riemannian manifolds with convex boundary, B.White proved the following alternative: Either there is an isoperimetric inequality for minimal hypersurfaces or there exists a closed minimal hypersurface, possibly with a small…

This is essentially an erratum, with some example to indicate inconsistencies. Suppose $A=k[X_1, X_2, \ldots, X_n]$ is a polynomial ring over a field $k$. The Complete Intersection conjecture states that, for any ideal $I$ in $A$,…

We prove that there are infinitely often pairs of primes much closer than the average spacing between primes - almost within the square root of the average spacing. We actually prove a more general result concerning the set of values taken…

In this paper we give new upper bounds on the regularity of edge ideals whose resolutions are k-steps linear; surprisingly, the bounds are logarithmic in the number of variables. We also give various bounds for the projective dimension of…

In this note we consider two simplicial arrangements of lines and ideals $I$ of intersection points of these lines. There are $127$ intersection points in both cases and the numbers $t_i$ of points lying on exactly $i$ configuration lines…

For a polynomial ring S in n variables, we consider the natural action of the symmetric group S_n on S by permuting the variables. For an S_n-invariant monomial ideal I in S and j >= 0, we give an explicit recipe for computing the modules…

Let S=K[x_1,x_2,...,x_n] be a polynomial ring in n variables over a field K. Stanley's conjecture holds for the modules I and S/I, when I is a critical monomial ideal. We calculate the Stanley depth of S/I when I is a canonical critical…

Given an ideal I in a polynomial ring, we consider the largest monomial subideal contained in I, denoted mono(I). We study mono as an interesting operation in its own right, guided by questions that arise from comparing the Betti tables of…

In this paper, we show for a monomial ideal $I$ of $K[x_1,x_2,\ldots,x_n]$ that the integral closure $\ol{I}$ is a monomial ideal of Borel type (Borel-fixed, strongly stable, lexsegment, or universal lexsegment respectively), if $I$ has the…