Related papers: Bounding Hilbert coefficients of parameter ideals

Let $h:\mathbb Z \to \mathbb Z_{\geq 0}$ be a nonzero function with $h(k)=0$ for $k\ll 0$. We define the Hilbert depth of $h$ by $\operatorname{hdepth}(h)=\max\{d\;:\; \sum_{j\leq k} (-1)^{k-j}\binom{d-j}{k-j}h(j)\geq 0\text{ for all }k\leq…

In this paper we study Cohen-Macaulay local rings of dimension $d$, multiplicity $e$ and second Hilbert coefficient $e_2$ in the case $e_2 = e_1 - e + 1$. Let $h = \mu(\mathfrak{m}) - d$. If $e_2 \neq 0$ then in our case we can prove that…

We had shown earlier that for a standard graded ring $R$ and a graded ideal $I$ in characteristic $p>0$, with $\ell(R/I) <\infty$, there exists a compactly supported continuous function $f_{R, I}$ whose Riemann integral is the HK…

A minimum depth d^I(S --> R) is assigned to a ring homomorphism S --> R and a R-R-bimodule I. The recent notion of depth of a subring d(S,R)in a paper by Boltje-Danz-Kuelshammer is recovered when I = R and S --> R is the inclusion mapping.…

In this paper we solve a problem posed by M.E. Rossi: {\it Is the Hilbert function of a Gorenstein local ring of dimension one not decreasing? } More precisely, for any integer $h>1$, $h \notin\{14+22k, \, 35+46k \ | \ k\in\mathbb{N} \}$,…

Let $I$ be a homogeneous ideal in the polynomial ring $R = k[z_1, \cdots, z_n]$ , where $k$ is an algebraically closed field of characteristic zero. Macaulay's Theorem provides constraints on the Hilbert function of $I$ or $R/I$ from one…

We determine a sharp lower bound for the Hilbert function in degree $d$ of a monomial algebra failing the weak Lefschetz property over a polynomial ring with $n$ variables and generated in degree $d$, for any $d\geq 2$ and $n\geq 3$. We…

Let $K$ be a field and let $R = K[X_1, \ldots, X_m]$ with $m \geq 2$. Give $R$ the standard grading. Let $I$ be a homogeneous ideal of height $g$. Assume $1 \leq g \leq m -1$. Suppose $H^i_I(R) \neq 0$ for some $i \geq 0$. We show (1)…

We study the normalization of a monomial ideal, and show how to compute its Hilbert function (using Ehrhart polynomials) if the ideal is zero dimensional. A positive lower bound for the second coefficient of the Hilbert polynomial is shown.

It is established a series of criteria for continuous and homeomorphic extension to the boundary of the so-called lower $Q$-homeomorphisms $f$ between domains in $\overline{\Rn}=\Rn\cup\{\infty\}$, $n\geqslant2$, under integral constraints…

The set of the first Hilbert coefficients of parameter ideals relative to a module--its Chern coefficients--over a local Noetherian ring codes for considerable information about its structure--noteworthy properties such as that of…

If a single particle obeys non-relativistic QM in R^d and has the Hamiltonian H = - Delta + f(r), where f(r)=sum_{i = 1}^{k}a_ir^{q_i}, 2\leq q_i < q_{i+1}, a_i \geq 0$, then the eigenvalues E = E_{n\ell}^{(d)}(\lambda) are given…

In [14] we introduced a new class of algebras, which we named \textit{quantum generalized Heisenberg algebras} and which depend on a parameter $q$ and two polynomials $f,g$. We have shown that this class includes all generalized Heisenberg…

Let $I$ be a perfect ideal of height two in $R=k[x_1, \ldots, x_d]$ and let $\varphi$ denote its Hilbert-Burch matrix. When $\varphi$ has linear entries, the algebraic structure of the Rees algebra $\mathcal{R}(I)$ is well-understood under…

Let $(S, \mathfrak n) $ be a regular local ring and let $I \subseteq \mathfrak n^2 $ be a perfect ideal of $S. $ Sharp upper bounds on the minimal number of generators of $I$ are known in terms of the Hilbert function of $R=S/I. $ Starting…

Let $(A,\mathfrak{m})$ be a hypersurface local ring of dimension $d \geq 1$, $N$ a perfect $A$-module and let $I$ be an ideal in $A$ with $\ell(N/IN)$ finite. We show that there is a integer $r_I \geq -1$ (depending only on $I$ and $N$)…

Let $M$ be a finitely generated module of dimension $d$ over a Noetherian local ring $(R,\m)$ and $\q $ the parameter ideal generated by a system of parameters $\x = (x_1,..., x_d)$ of $M$. For each positive integer $n$, set…

Let $I$ be an ideal of a local ring $(R,\mathfrak m)$ with $d = \dim R.$ For the local cohomology module $H^i_I(R)$ it is a well-known fact that it vanishes for $i > d$ and is an Artinian $R$-module for $i = d.$ In the case that the…

Let $(A,\mathfrak{m})$ be a regular local ring of dimension $d \geq 1$, $I$ an $\mathfrak{m}$-primary ideal. Let $N$ be a non-zero finitely generated $A$-module. Consider the functions \[ t^I(N, n) = \sum_{i = 0}^{ d}\ell(\text{Tor}^A_i(N,…

This paper explores the structure of quasi-socle ideals I=Q:m^2 in a Gorenstein local ring A, where Q is a parameter ideal and m is the maximal ideal in A. The purpose is to answer the problem of when Q is a reduction of I and when the…