Related papers: Asymptotic Multiplicities

We prove that limits of multiplicities associated to graded families of ideals exist under very general conditions. Most of our results hold for analytically unramified equicharacteristic local rings, with perfect residue fields. We give a…

We show the existence (and define) the mixed multiplicities of arbitrary graded families of ideals under mild assumptions. In particular, our methods and results are valid for the case of arbitrary $\mathfrak{m}$-primary graded families.…

This article focuses on the existence of asymptotic colengths for families of $\fm_{R}$-primary ideals in a Noetherian local ring $(R,\fm)$. In any characteristic, we generalize graded families to weakly graded families of ideals, and in…

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 $e_W(\mathfrak{I}):=\lim\limits_{n\to\infty}d!\frac{\ell_R(R/I_n)}{n^d}$…

The notion of $\varepsilon$-multiplicity was originally defined by Ulrich and Validashti as a limsup and they used it to detect integral dependence of modules. It is important to know if it can be realized as a limit. In this article we…

Let $R$ be a $d$-dimensional Noetherian local ring with maximal ideal $m_R$. In this article, we give a generalization of the multiplicity $e(I)$ of an $m_R$-primary ideal $I$ of $R$ to a multiplicity $e(\mathcal I)$ of a graded family of…

The notion of epsilon multiplicity was originally defined by Ulrich and Validashti as a limsup and they used it to detect integral dependence of modules. It is important to know if it can be realized as a limit. In this article we show that…

In an analytically unramified local ring $(R,\mathfrak m)$ of dimension $d\geq 1$, for a filtration of ideals $\mathfrak {I}=\{I_m\}_{m\in\mathbb N}$ satisfying $\mathfrak A(r)$ condition and for any $\mathfrak m$-primary ideal $K$, it is…

We find simple necessary and sufficient conditions on a local ring $R$ of dimension $d$ for the limit $$ \lim_{i\rightarrow\infty}\frac{\ell_R(R/I_n)}{n^d} $$ to exist whenever $\{I_n\}$ is a graded family of $m_R$-primary ideals, and give…

We develop a theory of multiplicities and mixed multiplicities of filtrations, extending the theory for filtrations of $m$-primary ideals to arbitrary (not necessarily Noetherian) filtrations. The mixed multiplicities of $r$ filtrations on…

In this exposition of the equality and inequality of Minkowski for multiplicity of ideals, we provide simple algebraic and geometric proofs. Connections with mixed multiplicities of ideals are explained.

In this paper, we initiate a systematic study of the generalized Hilbert-Kunz multiplicity for families of ideals in a Noetherian local ring (R,m) of positive characteristic, and introduce a new asymptotic invariant called the Amao-type…

We give simple necessary and sufficient conditions on projective schemes over a field k for asymptotic limits of the growth of all graded linear series of a fixed Kodaira-Iitaka dimension to exist. We also give necessary and sufficient…

We study deviation of ergodic averages for dynamical systems given by self-similar tilings on the plane and in higher dimensions. The main object of our paper is a special family of finitely-additive measures for our systems. An asymptotic…

In this work, we extend the concept of the double of an ideal defined in \cite{G2}, to the context of modules. We also obtain the genericity of the infinitesimal Lipschitz condition A for an enlarged class of analytic spaces.

We prove a multiplicity result for a class of strongly indefinite nonlinear second order asymptotically linear systems with Dirichlet boundary conditions. The key idea for the proof is to bring together the classical shooting method and the…

We prove an asymptotic analog of the classical Hurewicz theorem on mappings which lower dimension. This theorem allows us to find sharp upper bound estimates for the asymptotic dimension of groups acting on finite dimensional metric spaces…

Asymptotic properties of saturated powers of modules over a local domain R are studied. Under mild conditions, it is shown that the limit as k goes to infinity of the quotient of the saturation of the k-th power of a module E by the k-th…

On a Riemannian compact manifold, we give existence and multiplicity results for solutions of elliptic PDE by introducing isometry invariances. When the groups we used have finite orbits, we get multiplicity results for equations with the…

For a large integer $m,$ we obtain an asymptotic formula for the number of solutions of a certain congruence modulo $m$ with four variables, where the variables belong to special sets of residue classes modulo $m.$ This formula are applied…