Related papers: B\"ottcher coordinates

Let $X\subset \mathbb{C}^n; Y\subset \mathbb{C}^m$ be closed affine varieties and let $\phi: X\to Y$ be an algebraic bi-Lipschitz homeomorphism. Then ${\rm deg}\ X={\rm deg}\ Y.$ Similarly, let $(X,0)\subset (\mathbb{C}^n,0), (Y,0)\subset…

This note is devoted to two classical theorems: the open mapping theorem for analytic functions (OMT) and the fundamental theorem of algebra (FTA). We present a new proof of the first theorem, and then derive the second one by a simple…

In this paper, we equip a C*-algebra-valued b-metric spaces with a graph G = (V,E) and establish some common fixed point theorems. Also, some examples in support of our main results are provided. Finally, as applications, existence and…

The following pullback problem will be considered. Given a finite holomorphic map germ $\phi : (\mathbb{C}^{n}, 0) \to (\mathbb{C}^{n}, 0)$ and an analytic germ $X$ in the target, if the preimage $Y = \phi^{-1}(X)$, taken with the reduced…

We prove that any complex or real analytic set or function germ is topologically equivalent to a germ defined by polynomial equations whose coefficients are algebraic numbers.

In this note, continuous transitive maps $f$ on the interval $I$ are re-addressed, where $I$ denotes one of the intervals: $(-\infty, \infty)$, $(-\infty, a]$, $[b, \infty)$, $[a, b]$, where $a < b$ are real numbers. Such maps must have a…

We study conjugacy classes of germs of non-flat diffeomorphisms of the real line fixing the origin. Based on the work of Takens and Yoccoz, we establish results that are sharp in terms of differentiability classes and order of tangency to…

Let X be an analytic vector field on a real or complex 2-manifold, and K a compact set of zeros of X whose fixed point index is not zero. Let A denote the Lie algebra of analytic vector fields Y on M such that at every point of M the values…

A classification theorem is obtained for a class of unital simple separable amenable Z-stable C*-algebras which exhausts all possible values of the Elliott invariant for unital stably finite simple separable amenable Z-stable C*-algebras.…

We investigate the local dynamics of antiholomorphic diffeomorphisms around a parabolic fixed point. We first give a normal form. Then we give a complete classification including a modulus space for antiholomorphic germs with a parabolic…

We consider graphs E which have been obtained by adding one or more sinks to a fixed directed graph G. We classify the C*-algebra of E up to a very strong equivalence relation, which insists, loosely speaking, that C*(G) is kept fixed. The…

In this short note we prove a version of Bertini's theorem for unipotent rigid fundamental groups, stating that for every smooth, projective, geometrically connected variety $X$ over an infinite perfect field $k$ of characteristic $p>0$,…

Zilber's Exponential Algebraic Closedness conjecture (also known as Zilber's Nullstellensatz) gives conditions under which a complex algebraic variety should intersect the graph of the exponential map of a semiabelian variety. We prove the…

We consider the Riemann Mapping Theorem in the case of a bounded simply connected and semianalytic domain. We show that the germ at 0 of the Riemann map (i.e. biholomorphic map) from the upper half plane to such a domain can be realized in…

Let $ f$ be a real-analytic function germ whose critical locus contains a given real-analytic set $ X $, and let $ Y $ be a germ of closed subset of $ \mathbb{R}^n $ at the origin. We study the stability of $ f $ under perturbations $ u $…

In this paper, we prove Fukui-Kurdyka-Paunescu's Conjecture, which says that subanalytic arc-analytic bi-Lipschitz homeomorphisms preserve the multiplicities of real analytic sets. We also prove several other results on the invariance of…

Let F be a germ of (C^2,O) tangent to the identity. Assume F has a characteristic direction [v]. In [Hak] Hakim gives conditions to guarantee the existence of an attracting basin to the origin along [v], in the case of [v] a non-degenerate…

In this paper we give a complete description of K-theory groups for Cuntz-Krieger C*-algebras associated to general locally-finite (topologically connected) graphs via Bass-Hashimoto operator. Our result generalizes the one obtained by the…

We prove the multisummability of the infinitesimal generator of unfoldings of finite codimension tangent to the identity 1-dimensional local complex analytic diffeomorphisms. We also prove the multisummability of Fatou coordinates and…

Let $F:(\mathbb{C}^2,0)\to (\mathbb{C}^n,0)$ be the germ of a finite map and $(X,0)$ be its image. We will in this article using the topology of the link show that $(X,0)$ has to be a quotient singularity if it is normal and describe the…