Related papers: Normalization of Ideals and Briancon-Skoda Numbers

Let $R=K[x_1,\ldots, x_n]$ be the polynomial ring in $n$ variables over a field $K$ and let $M_{n,t}=(x^{e_1},\ldots, x^{e_n})$ be a monomial ideal of $R$, where $x^{e_i}=x_1^t\ldots x_{i-1}^tx_{i+1}^t\ldots x_n^t$. We study the unmixedness…

The Hilbert scheme of $n$ points in the affine plane contains the open subscheme parametrizing $n$ distinct points in the affine plane, and the closed subscheme parametrizing ideals of codimension $n$ supported at the origin of the affine…

An approach to (normalized) infinite dimensional integrals, including normalized oscillatory integrals, through a sequence of evaluations in the spirit of the Monte Carlo method for probability measures is proposed. in this approach the…

Let $S = K[x_1, \ldots, x_n]$ denote the polynomial ring in $n$ variables over a field $K$ with each $\deg x_i = 1$ and $I \subset S$ a homogeneous ideal of $S$ with $\dim S/I = d$. The Hilbert series of $S/I$ is of the form…

This paper provides an extensive study of the $\mathscr{I}$-Miller null ideals $M_\mathscr{I}$, $\sigma$-ideals on the Baire space parametrized by ideals $\mathscr{I}$ on countable sets. These $\sigma$-ideals are associated to the idealized…

To a multi-index filtration (say, on the ring of germs of functions on a germ of a complex analytic variety) one associates several invariants: the Hilbert function, the Poincar\'e series, the generalized Poincar\'e series, and the…

Equality of the second order arithmetic means of two principal ideals does not imply equality of their first order arithmetic means (second order equality cancellation). We provide fairly broad sufficient conditions on one of the principal…

We design observer-based controllers to stabilise abstract linear boundary control systems on Hilbert spaces. Our main results introduce conditions for exponential, strong, and polynomial stability, and establish external well-posedness of…

In this article, we extend the notion of multiplicity for weakly graded families of ideals which are bounded below linearly. In particular, we show that the limit $e_W(\mathfrak{I}):=\lim\limits_{n\to\infty}d!\frac{\ell_R(R/I_n)}{n^d}$…

Let I = (F_1,...,F_r) be a homogeneous ideal of R = k[x_0,...,x_n] generated by a regular sequence of type (d_1,...,d_r). We give an elementary proof for an explicit description of the graded Betti numbers of I^s for any s \geq 1. These…

Let Hilb^p be the Hilbert scheme parametrizing the closed subschemes of P^n with Hilbert polynomial p\in Q[t] over a field K of characteristic zero. By bounding below the cohomological Hilbert functions of the points of Hilb^p we define…

By introducing Hilbert space and operators, we show how probabilities, approximations and entropy encoding from signal and image processing allow precise formulas and quantitative estimates. Our main results yield orthogonal bases which…

It is a well-known conjecture, sometimes attributed to Frankl, that for any family of sets which is closed under the union operation, there is some element which is contained in at least half of the sets. Gilmer was the first to prove a…

The Hilbert function of a module over a positively graded algebra is of quasi-polynomial type (Hilbert--Serre). We derive an upper bound for its grade, i.e. the index from which on its coefficients are constant. As an application, we give a…

We identify necessary and sufficient conditions on $k$th order differential operators $\mathbb{A}$ in terms of a fixed halfspace $H^+\subset\mathbb{R}^n$ such that the Gagliardo--Nirenberg--Sobolev inequality $$…

We study the recovery of multivariate functions from reproducing kernel Hilbert spaces in the uniform norm. Our main interest is to obtain preasymptotic estimates for the corresponding sampling numbers. We obtain results in terms of the…

Let $R^h$ denote the polynomial ring in variables $x_1,\,\ldots,\, x_h$ over a specified field $K$. We consider all of these rings simultaneously, and in each use lexicographic (lex) monomial order with $x_1 > \cdots > x_h$. Given a fixed…

Let $(R,m)$ be a Noetherian local ring of dimension $d$ and $K,Q$ be $m$-primary ideals in $R.$ In this paper we study the finiteness properties of the sets $\Lambda_i^K(R):=\{g_i^K(Q): Q$ is a parameter ideal of $R\},$ where $g_i^K(Q)$…

We consider general second order uniformly elliptic operators subject to homogeneous boundary conditions on open sets $\phi (\Omega)$ parametrized by Lipschitz homeomorphisms $\phi $ defined on a fixed reference domain $\Omega$. Given two…

Let $J\subset I$ be ideals in a formally equidimensional local ring with $\lambda(I/J)<\infty.$ Rees proved that for all $n\gg0$, $\lambda(I^n/J^n)$ is a polynomial $P(I/J)(X)$ in $n$ of degree at most dim $R$ and $J$ is a reduction of $I$…