Related papers: Interpolating Between Hilbert-Samuel and Hilbert-K…

This paper gives the recursion formula for mixed multiplicities of maximal degrees with respect to joint reductions of ideals, which is one of important results in the mixed multiplicity theory. Using this result, we give consequences on…

This paper develops a theory of equimultiplicity for Hilbert-Kunz multiplicity and uses it to study the behavior of Hilbert-Kunz multiplicity on the Brenner-Monsky hypersurface. A number of applications follows, in particular we show that…

Let S be a finitely generated standard multi-graded algebra over a Noetherian local ring A. This paper first expresses mixed multiplicities of S in term of Hilbert-Samuel multiplicity that explained the mixed multiplicities S as the…

For a pair $(M, I)$, where $M$ is finitely generated graded module over a standard graded ring $R$ of dimension $d$, and $I$ is a graded ideal with $\ell(R/I) < \infty$, we introduce a new invariant $HKd(M, I)$ called the {\em Hilbert-Kunz…

We prove that, under certain assumptions, generalized Hilbert-Kunz multiplicities can be expressed as linear combinations of classical Hilbert-Kunz multiplicities.

The main purpose of this paper is to provide formulas for the Hilbert-Kunz multiplicity of fiber product rings and Nagata idealizations. We give explicit formulas for the Hilbert-Kunz multiplicity of a fiber product $R \times_T S$, where…

The j-multiplicity is an invariant that can be defined for any ideal in a Noetherian local ring $(R, m)$. It is equal to the Hilbert-Samuel multiplicity if the ideal is $m$-primary. In this paper we explore the computability of the…

The notion of regularity has been used by S. Kleiman in the construction of bounded families of ideals or sheaves with given Hilbert polynomial, a crucial point in the construction of Hilbert or Picard scheme. In a related direction,…

Let R denote a two-dimensional normal standard-graded domain over the algebraic closure K of a finite field of characteristic p, and let I denote a homogeneous primary ideal. We prove that the Hilbert-Kunz function of I has the form =…

Given ideals $I,J$ of a noetherian local ring $(R, \mathfrak m)$ such that $I+J$ is $\mathfrak m$-primary and a finitely generated $R$-module $M$, we associate an invariant of $(M,R,I,J)$ called the $h$-function. Our results on…

This paper focuses on a numerical invariant for local rings of characteristic $p$ called $h$-function, that recovers several important invariants, including the Hilbert-Kunz multiplicity, $F$-signature, $F$-threshold, and $F$-signature of…

In this paper we prove that the Watanabe-Yoshida conjecture holds up to dimension $7$. Our primary new tool is a function, $\varphi_J\left(R; z^t\right),$ that interpolates between the Hilbert-Kunz multiplicities of a base ring, $R$, and…

We present a new approach to the study of multiplier ideals in a local, two-dimensional setting. Our method allows us to deal with ideals, graded systems of ideals and plurisubharmonic functions in a unified way. Among the applications are…

We generalize the notion of Hilbert-Kunz multiplicity of a graded triple $(M,R,I)$ in characteristic $p>0$ by proving that for any complex number $y$, the limit $$\underset{n \to \infty}{\lim}(\frac{1}{p^n})^{\text{dim}(M)}\sum \limits_{j=…

Each symmetrically-normed ideal $\mathcal{I}$ of compact operators on a Hilbert space $H$ induces a multiplier topology $\mu^*_{\mathcal{I}}$ on the algebra $\mathcal{B}(H)$ of bounded operators. We show that under fairly reasonable…

We introduce the notion of a pseudomultiplier of a Hilbert space $\mathcal H$ of functions on a set $\Omega$. Roughly, a pseudomultiplier of $\mathcal H$ is a function which multiplies a finite-codimensional subspace of $\mathcal H$ into…

We explore connections between the generalized multiplicities of square-free monomial ideals and the combinatorial structure of the underlying hypergraphs using methods of commutative algebra and polyhedral geometry. For instance, we show…

Let p be a prime. The Hilbert-Kunz multiplicity, mu, of the element sum(x_i^(d_i)) of (Z/p)[x_1,..., x_s] depends on p in a complicated way. We calculate the limit of mu as p -> infinity. In particular when each d_i is 2 we show that the…

In this article we give explicit descriptions of the multiplicities of some classes of monomial ideals. For instance, we give a formula for the multiplicities of all codimension 1 monomial ideals, and another formula for the multiplicities…

We show that ideal submodules and closed ternary ideals in Hilbert modules are the same. We use this insight as a little peg on which to hang a little note about interrelations with other notions regarding Hilbert modules. In Section 3, we…