Related papers: D\'etermination finie sur un espace de Stein

We introduce the concept of topological finite-determinacy for germs of analytic functions within a fixed ideal $I$, which provides a notion of topological finite-determinacy of functions with non-isolated singularities. We prove the…

The Tjurina ideal of a germ of an holomorphic function $f$ is the ideal of $\mathscr{O}_{\mathbbm{C}^n,0}$ - the ring of those germs at $0\in\mathbbm{C}^n$ - generated by $f$ itself and by its partial derivatives. Here it is denoted by…

Let $R=K[[x_1,...,x_s]]$ be the ring of formal power series with maximal ideal $\mathfrak{m}$ over a field $K$ of arbitrary characteristic. On the ring $M_{m,n}$ of $m\times n$ matrices $A$ with entries in $R$ we consider several…

We suggest the necessary/sufficient criteria for the existence of a (order-by-order) solution y(x) of a functional equation F(x,y)=0 over a ring. In full generality, the criteria hold in the category of filtered groups, this includes the…

Our main result is to give necessary and sufficient conditions, in terms of Fourier transforms, on a closed ideal $I$ in $\loneg$, the space of radial integrable functions on $G=SU(1,1)$, so that $I=\loneg$ or $I=\lonez$---the ideal of…

We extend the circle of ideas from a previous paper on hypersurfaces to functions $f \colon (\mathbb C^n, 0) \to (\mathbb C^k, 0)$ with an isolated singularity in a stratified sense on an arbitrary, but fixed complex analytic germ $(X, 0)$.…

Topologies on algebraic and equational theories are used to define germ determined, near-point determined, and point determined rings of smooth functions, without requiring them to be finitely generated. It is proved, that any commutative…

We give the definition of the Thom condition and we show that given any germ of complex analytic function $f:(X,x)\to(\mathbb{C},0)$ on a complex analytic space $X$, there exists a geometric local monodromy without fixed points, provided…

Fix a module M over a local ring R and a group action G on M, not necessarily R-linear. To understand how large is the G-orbit of an element z\in M one looks for the large submodules of M lying in Gz. We provide the corresponding…

We give necessary and sufficient conditions of infinite determinacy for smooth function germs whose critical locus contains a given set. This set is assumed to be the zero variety X of some analytic map germ having maximal rank on a dense…

In this paper we prove an analogue of the discrete spherical maximal theorem of Magyar, Stein, and Wainger, an analogue which concerns maximal functions associated to homogenous algebraic surfaces. Let $\mathfrak{p}$ be a homogenous…

We consider matrices with entries in a local ring, Mat(m,n,R). Fix a group action, G on Mat(m,n,R), and a subset of allowed deformations, \Sigma\subseteq Mat(m,n,R). The standard question of Singularity Theory is the…

Systems of germs of sets in infinite-dimensional spaces are introduced and studied. Such a system corresponds to a local zero-set of an ideal of the ring of analytic functions of infinite number of variables. Conversely, this system of…

Let $G$ be a reductive group over a local field $F$ satisfying the assumptions of \cite{Deb1}, $G_{reg}\subset G$ the subset of regular elements. Let $T\subset G$ be a maximal torus. We write $T_{reg}=T\cap G_{reg}$. Let $dg ,dt$ be Haar…

We prove that every map-germ ${f \bar g}: (\C^n,\0) {\to}(\C,0)$ with an isolated critical value at 0 has the Thom $a_{f \bar g}$-property. This extends Hironaka's theorem for holomorphic mappings to the case of map-germs $f \bar g$ and it…

The goal of this article is to establish tauberian theorems for the $k$--summability processes defined by germs of analytic functions in several complex variables. The proofs are based on the tauberian theorems for $k$--summability in one…

This paper investigates continuity properties of value functions and solutions for parametric optimization problems. These problems are important in operations research, control, and economics because optimality equations are their…

It is shown in a local strongly $F$-regular ring there exits natural number $e_0$ so that if $M$ is any finitely generated maximal Cohen-Macaulay module then the pushforward of $M$ under the $e_0$th iterate of the Frobenius endomorphism…

Let $G$ be a group. We say that an element $f\in G$ is {\em reversible in} $G$ if it is conjugate to its inverse, i.e. there exists $g\in G$ such that $g^{-1}fg=f^{-1}$. We denote the set of reversible elements by $R(G)$. For $f\in G$, we…

Let $\mathcal{E}_n$ be the ring of the germs of $\mathcal{C}^\infty$-functions at the origin in $\R^n$. It is well known that if $I$ is an ideal of $\mathcal{E}_n$, generated by a finite number of germs of analytic functions, then $I$ is…