Related papers: A function determined by a hypersurface of positiv…

Suppose that h in F[x,y,z], char F=2, defines a nodal cubic. In earlier papers we made a precise conjecture as to the Hilbert-Kunz functions attached to the powers of h. Assuming this conjecture we showed that a class of characteristic 2…

The main goal of this paper is to prove, in positive characteristic $p$, stability behavior for the graded Betti numbers in the periodic tails of the minimal resolutions of Frobenius powers of the homogeneous maximal ideals for very general…

Let $k$ be a field of characteristic $\ne 2$ and let $Q_{n,m}(x_1, ...,x_n,y_1, ...,y_m)=x_1^2+ ... +x_n^2-(y_1^2+ ... +y_m^2)$ be a quadratic form over $k$. Let $R(Q_{n,m})=R_{n,m}=k[x_1, ...,x_n,y_1, ...,y_m]/(Q_{n,m}-1)$. In this note we…

This article presents the construction of a non-affine hypersurface on an $n$-simplex in $\mathbb{R}^n$. Additionally, fractal dimension of the graph of a non-affine multivariate real-valued fractal function is estimated under certain…

In this paper we analyse the structure of the spaces of smooth type functions, generated by elements of arbitrary Hilbert spaces, as a continuation of the research in our previous papers in this series. We prove that these spaces are…

We investigate the defining ideal of a set of points X in multi-projective space with a special emphasis on the case that X is in generic position, that is, X has the maximal Hilbert function. When X is in generic position, we determine the…

Let $L$ be a homogeneous divergence form higher order elliptic operator with complex bounded measurable coefficients and $(p_-(L),\, p_+(L))$ be the maximal interval of exponents $q\in[1,\,\infty]$ such that the semigroup…

In this note we study one-dimensional definable sets in power series fields with perfect residue fields. Using the description of automorphisms given by Schilling, in \cite{S44}, we show that such sets are unions of existentially definable…

Let $f = f(z,t)$ be a function holomorphic in $z \in O \subseteq {\mathbb C}^d$ for fixed $t\in \Omega$ and measurable in $t$ for fixed $z$ and such that$z \mapsto f(z,\cdot)$ is bounded with values in$E := L_{p}(\Omega)$, $1\le p \le…

In this paper, we prove that transnormal functions are isoparametric functions on Finsler space forms (N(c), F) under certain conditions, which generalize Theorem B given by Q.M. Wang in Riemannian case. Next, we discuss the relationship…

Let $k[x]_{(x)}$ be the polynomial ring $k[x]$ localized in the maximal ideal $(x)\subseteq k[x]$. We study the Hilbert functor parameterizing ideals of colength $n$ in this ring {\it having support at the origin}. The main result of this…

Let X/S be a quasi-projective morphism over an affine base. We develop in this article a technique for proving the existence of closed subschemes H/S of X/S with various favorable properties. We offer several applications of this technique,…

Let $X\subset{\mathbb R}^n$ be a (global) real analytic surface. Then every positive semidefinite meromorphic function on $X$ is a sum of $10$ squares of meromorphic functions on $X$. As a consequence, we provide a real Nullstellensatz for…

Let X be a finite set of points in R^n. A polynomial p nonnegative on X can be written as a sum of squares of rational functions modulo the vanishing ideal I(X). From the point of view of applications, such as polynomial optimization, we…

A characterization of valuations on the space of convex Lipschitz functions whose domain is a polytope in $\mathbb{R}^n$ is obtained. It is shown that every upper semicontinuous, equi-affine and dually epi-translation invariant valuation…

Let $\mathbb{F}_q$ be a finite field with $q$ elements, $\psi$ a non-zero element of $\mathbb{F}_q$, and $n$ an integer $\geq 3$ prime to $q$. The aim of this article is to show that the zeta function of the projective variety over…

Let $\mathbb Q_{\epsilon_i}^{n_i}$ denote the simply connected space form of dimension $n_i\ge 2$ and constant sectional curvature $\epsilon_i$. We prove that any connected isoparametric hypersurface of $\mathbb…

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 aim of the present paper is the study of some classes of real hypersurfaces equipped with the condition \phi l = l \phi, (l = R(., \xi, \xi))

We establish the continuity of Hilbert-Kunz multiplicity and F-signature as functions from a Cohen-Macaulay local ring $(R,\m,k)$ of prime characteristic to the real numbers at reduced parameter elements with respect to the $\m$-adic…