Related papers: Analysis in Hilbert-Kunz theory

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…

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…

The aim of this manuscript is to discuss the Hilbert-Kunz functions over an excellent local ring regular in codimension one. We study the shape of the Hilbert-Kunz functions of modules and discuss the properties of the coefficient of the…

Given a commutative local ring $(R,\mathfrak m)$ and an ideal $I$ of $R$, a family of quotients of the Rees algebra $R[It]$ has been recently studied as a unified approach to the Nagata's idealization and the amalgamated duplication and as…

Suppose R is a Noetherian local ring with prime characteristic p>0. In this article, we show the existence of a local numerical invariant, called the F-signature, which roughly characterizes the asymptotic growth of the number of splittings…

The affine Hilbert function is a classical algebraic object that has been central, among other tools, to the development of the polynomial method in combinatorics. Owing to its concrete connections with Gr\"obner basis theory, as well as…

We use the results of our paper "p-Fractals and power series--I" (Journal of Algebra 280, 2004, pp. 505--536) to prove the rationality of the Hilbert-Kunz series of a large family of power series, including those of the form \sum_i…

Suppose B=F[x,y,z]/h is the homogeneous coordinate ring of a characteristic p degree 3 irreducible plane curve C with a node. Let J be a homogeneous (x,y,z)-primary ideal and n -> e_n be the Hilbert-Kunz function of B with respect to J. Let…

We continue the study of intersection algebras $\mathcal B = \mathcal B_R(I, J)$ of two ideals $I, J$ in a commutative Noetherian ring $R$. In particular, we exploit the semigroup ring and toric structures in order to calculate various…

We show that the Hilbert-Kunz multiplicity of a $d$-dimensional nonregular complete intersection over the algebraic closure of $F_p$, $p>2$ prime, is bounded by below by the Hilbert-Kunz multiplicity of the hypersurface $\sum _{i=0}^{d}…

The classical Hilbert specialization property is a field-theoretic tool ensuring that polynomial irreducibility over a field is preserved under specialization of some of the variables. We develop an integral counterpart by introducing the…

A recent continuous family of multiplicity functions on local rings was introduced by Taylor interpolating between Hilbert-Samuel and Hilbert-Kunz multiplicities. The obvious goal is to use this as a tool for deforming results from one to…

After introducing the sub-Riemannian geometry of the Heisenberg group Hn, n \geq 1, we recall some basics about hypersurfaces endowed with the H-perimeter measure and horizontal Green's formulas. Then, we describe a class of compact closed…

We determine the Hilbert-Kunz function of plane elliptic curves in odd characteristic, as well as over arbitrary fields the generalized Hilbert-Kunz functions of nodal cubic curves. Together with results of K. Pardue and P. Monsky, this…

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,…

Here we prove that the Hilbert-Kunz mulitiplicity of a quadric hypersurface of dimension $d$ and odd characteristic $p\geq 2d-4$ is bounded below by $1+m_d$, where $m_d$ is the $d^{th}$ coefficient in the expansion of…

The paper investigates the behavior of Hilbert-Samuel and Hilbert-Kunz multiplicities in families of ideals. It is shown that Hilbert-Samuel multiplicity is upper semicontinuous almost generally and that Hilbert-Kunz multiplicity is upper…

Given the growing quantity of proposals and works of basic hypergeometric functions in the scope of $q$-calculus, it is important to introduce a systematic classification of $q$-calculus. Our aim in this article is to investigate certain…

Here we compute Hilbert-Kunz functions of any nontrivial ruled surface over ${\bf P}^1_k$, with respect to all ample line bundles on it.

Let $H$ be a hyperexponential function in $n$ variables $x=(x_1,\dots,x_n)$ with coefficients in a field $\mathbb{K}$, $[\mathbb{K}:\mathbb{Q}] <\infty$, and $\omega$ a rational differential $1$-form. Assume that $H\omega$ is closed and $H$…