Related papers: Sur quelques aspects de la g\'eom\'etrie de l'espa…

In the first half of twentieth century the theory of complex analytic functions and of their zerosets was fully developed. The definition of holomorphic function has a local nature. Germs of holomorphic functions form a distinguished…

We prove surjectivity result in Fr\'echet spaces of Nash-Moser type. That is, with uniform estimates over all semimorms. Our method works for functions which are only continuous and G\^ateaux differentiable like in the recent result of…

Let (S,0) be a germ of complex analytic normal surface. On its minimal resolution, we consider the reduced exceptional divisor E and its irreducible components E_i. The Nash map associates to each irreducible component C_k of the space of…

We study complex Lie algebras spanned by pairs \left(Z,Y\right) of germs of a meromorphic vector field of the complex plane satisfying \left[Z,Y\right]=\delta Y for some \delta\in\ww C . This topic relates to Liouville-integrability of the…

We characterize all $f$-algebra products on AM-spaces by constructing a canonical AM-space $W_X$ associated to each AM-space $X$, such that the $f$-algebra products on $X$ correspond bijectively to the positive cone $(W_X)_+$. This…

The Nash multiplicity sequence was defined by M. Lejeune-Jalabert as a non-increasing sequence of integers attached to a germ of a curve inside a germ of a hypersurface. M. Hickel generalized this notion and described a sequence of blow ups…

In this article we study the dynamics generated by germs of parabolic diffeomorphisms f : (C; 0)->(C; 0) tangent to the identity. We show how formal classification of a given parabolic diffeomorphism can be deduced from the asymptotic…

Let $(f,g): (S,s) \to (\mathbb{C}^2, 0)$ be a finite morphism from a germ of normal complex analytic surface to the germ of $\mathbb{C}^2$ at the origin. We show that the affine algebraic curve in $\mathbb{C}^2$ defined by the initial…

Let $\xi$ be an analytic bracket-generating distribution. We show that the subspace of germs that are singular (in the sense of Control Theory) has infinite codimension within the space of germs of smooth curves tangent to $\xi$. We…

Category theory provides a collective description of many arrangements in mathematics, such as topological spaces, Banach spaces and game theory. Within this collective description, the perspective from any individual member of the…

Let $f(x,y), g(x,y)$ denote either a pair of holomorphic function germs, or a pair of monic polynomials in $x$ whose coefficients are Laurent series in $y$. A relative polar arc is a Newton-Puiseux root, $x=\gamma(y)$, of the Jacobian…

We combine several mini miracles to achieve an elementary understanding of infinite loop spaces and very effective spectra in the algebro-geometric setting of motivic homotopy theory. Our approach combines $\Gamma$-spaces and framed…

We prove the equivalence between geometric and analytic definitions of quasiconformality for a homeomorphism $f\colon X\rightarrow Y$ between arbitrary locally finite separable metric measure spaces, assuming no metric hypotheses on either…

Let $X$ and $Y$ be pseudocompact spaces and let the function $\Phi: X\times Y\to \mathbb R$ be separately continuous. The following conditions are equivalent: (1) there is a dense $G_\delta$ subset of $D\subset Y$ so that $\Phi$ is…

Let X be a separable Banach space which admits a separating polynomial; in particular X a separable Hilbert space. Let $f:X \rightarrow R$ be bounded, Lipschitz, and $C^1$ with uniformly continuous derivative. Then for each {\epsilon}>0,…

We consider a finite analytic morphism $\phi = (f,g) : (X,p)\to (\C^2,0)$ where $(X,p)$ is a complex analytic normal surface germ and $f$ and $g$ are complex analytic function germs. Let $\pi : (Y,E_{Y})\to (X,p)$ be a good resolution of…

Using nonstandard analysis we define a topology on the ring of germs of functions: $(mathbb R^n,0)\rightarrow(mathbb R,0)$. We prove that this topology is absolutely convex, Hausdorff, that convergent nets of continuous germs have…

We show how formal and rigid geometry can be used in the theory of complex singularities, and in particular in the study of the Milnor fibration and the motivic zeta function. We introduce the so-called analytic Milnor fiber associated to…

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…

In this article we prove that every germ of analytic meromorphic function at $(\mathbb{C}^{2},0)$ is equivalent, under the right composition by a germ of biholomorphism, to a germ of algebraic meromorphic function. An analogous result is…