Related papers: Hilbert-Kunz functions over rings regular in codim…

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…

We study the behavior of the Hilbert-Kunz multiplicity of powers of an ideal in a local ring. In dimension two, we provide answers to some problems raised by Smirnov, and give a criterion to answer one of his questions in terms of a…

In singularity theory or algebraic geometry, it is natural to investigate possible Hilbert functions for special algebras $A$ such as local complete intersections or more generally Gorenstein algebras. The sequences that occur as {the}…

We prove that, analogous to the HK density function, (used for studying the Hilbert-Kunz multiplicity, the leading coefficient of the HK function), there exists a $\beta$-density function $g_{R, {\bf m}}:[0,\infty)\longrightarrow {\mathbb…

In this note, we provide several characterizations of regular local rings in positive characteristics, in terms of the Hilbert-Kunz multiplicity and its higher $\tor$ counterparts $\i t_i=\underset{n \to \infty}{\lim} \l(\tor_i(k,{}^{f^n}…

Let R be a polynomial ring and M a finitely generated graded R-module of maximal grade (which means that the ideal I_t(\cA) generated by the maximal minors of a homogeneous presentation matrix, \cA, of M has maximal codimension in R).…

In this article, I give an iterative closed form formula for the Hilbert-Kunz function for any binomial hypersurface in general, over any feild of arbitrary positive characteristic. I prove that the Hilbert-Kunz multiplicity associated to…

Let $(A,\mathfrak{m})$ be a complete intersection ring of dimension $d$ and let $I$ be an $\mathfrak{m}$-primary ideal. Let $M$ be a maximal \CM \ $A$-module. For $i = 0,1,\cdots,d$, let $e_i^I(M)$ denote the $i^{th}$ Hilbert -coefficient…

Let $k$ be a field and $x,y$ indeterminates over $k$. Let $R=k[x^a,x^{p_1}y^{s_1},\ldots,x^{p_t}y^{s_t},y^b] \subseteq k[x,y]$. We calculate the Hilbert polynomial of $(x^a,y^b)$. The multiplicity of this ideal provides part of a criterion…

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 prove that the Hilbert-Kunz multiplicity is upper semi-continuous in F-finite rings and algebras of essentially finite type over an excellent local ring.

We present results on the Watanabe-Yoshida conjecture for the Hilbert-Kunz multiplicity of a local ring of positive characteristic. By improving on a "volume estimate" giving a lower bound for Hilbert-Kunz multiplicity, we obtain the…

This first part of the paper describes the support of top graded local cohomology modules. As a corrolary one obtains a simple criteria for the vanishing of these modules and also the fact that they have finitely many minimal primes. The…

In this presentation we shall deal with some aspects of the theory of Hilbert functions of modules over local rings, and we intend to guide the reader along one of the possible routes through the last three decades of progress in this area…

For a standard graded ring $R$ of dimension $\geq 2$ over a perfect field of characteristic $p>0$ and a homogeneous ideal $I$ of finite colength, the HK density function of $R$ with respect to $I$ is a compactly supported continuous…

Let $(R,\m)$ be a formally unmixed local ring of positive prime characteristic and dimension $d$. We examine the implications of having small Hilbert-Kunz multiplicity (i.e., close to 1). In particular, we show that if $R$ is not regular,…

Let R be a polynomial ring in r variables and D a dual ring upon which R acts as partial differential operators (classical apolarity). For a type two graded level Artinian algebras A=R/I, of socle degree j we consider the family of Artinian…

Let $O$ be a one-dimensional Cohen-Macaulay local ring having a finite field as a coefficient field. The aim of this work is to extend the explicit computations of the St\"ohr Zeta Function of $O$ for one and two branches to an arbitrary…

Let $(R,\frak{m})$ be a $d$-dimensional Cohen-Macaulay local ring, $I$ an $\frak{m}$-primary ideal and $J$ a minimal reduction of $I$. In this paper we study the independence of reduction ideals and the behavior of the higher Hilbert…

We study Hilbert-Kunz multiplicity of the powers of an ideal and establish existence of the second coefficient at the full level of generality, thus extending a recent result of Trivedi. We describe the second coefficient as the limit of…