Germs Of Analytic Functions Research Articles

Arc spaces, motivic measure and Lipschitz geometry of real algebraic sets

We investigate connections between Lipschitz geometry of real algebraic varieties and properties of their arc spaces. For this purpose we develop motivic integration in the real algebraic set-up. We construct a motivic measure on the space of real analytic arcs. We use this measure to define a real motivic integral which admits a change of variables formula not only for the birational but also for generically one-to-one Nash maps. As a consequence we obtain an inverse mapping theorem which holds for continuous rational maps and, more generally, for generically arc-analytic maps. These maps appeared recently in the classification of singularities of real analytic function germs. Finally, as an application, we characterize in terms of the motivic measure, germs of arc-analytic homeomorphism between real algebraic varieties which are bi-Lipschitz for the inner metric.

Asymptotic expansions and summability with respect to an analytic germ

In a previous article [CMS], monomial asymptotic expansions, Gevrey asymptotic expansions, and monomial summability were introduced and applied to certain systems of singularly perturbed differential equations. In the present work, we extend this concept, introducing (Gevrey) asymptotic expansions and summability with respect to a germ of an analytic function in several variables – this includes polynomials. The reduction theory of singularities of curves and monomialization of germs of analytic functions are crucial to establish properties of the new notions, for example a generalization of the Ramis–Sibuya theorem for the existence of Gevrey asymptotic expansions. Two examples of singular differential equations are presented for which the formal solutions are shown to be summable with respect to a polynomial: one ordinary and one partial differential equation.

Moduli spaces of a family of topologically non quasi-homogeneous functions

We consider a topological class of a germ of complex analytic function in two variables which does not belong to its jacobian ideal. Such a function is not quasi homogeneous. Each element f in this class induces a germ of foliation (df = 0). Proceeding similarly to the homogeneous case [2] and the quasi homogeneous case [3] treated by Genzmer and Paul, we describe the local moduli space of the foliations in this class and give analytic normal forms. We prove also the uniqueness of these normal forms.

Vanishing zones and the topology of non-isolated singularities

We compare the topology of the link $$L_0$$ of non-isolated singularities defined by real analytic map-germs $$({\mathbb {R}}^m,0) \buildrel {h} \over {\rightarrow } ({\mathbb {R}}^n,0)$$ , $$m > n$$ , with that of the boundary of a local non-critical level of h. We show that if the germ of h has an isolated critical value at $$0 \in {\mathbb {R}}^n$$ and admits a local Milnor-Le fibration at 0, then there exists “a vanishing zone for h”. This is an appropriate neighborhood of the set $$L_0 \cap \Sigma $$ , where $$\Sigma $$ denotes the critical set of h, such that away from it the topology of $$L_0$$ is fully determined by the boundary of the corresponding local Milnor fibre. We give conditions for the vanishing zone to be a fiber bundle over $$L_0 \cap \Sigma $$ . A particular class of real singularities we envisage in this paper are those of the type $$f\bar{g}: ({\mathbb {C}}^n,0) \rightarrow ({\mathbb {C}},0)$$ with f, g holomorphic and satisfying certain conditions. We introduce for these a regularity criterium for having a local Milnor-Le fibration, and we use this to produce an example of a real analytic singularity which does not have the Thom $$a_f$$ -property and yet has a local Milnor-Le fibration. Throughout this work we provide explicit examples of functions satisfying the hypothesis we need in each section.

Strong accessibility and integral manifolds of the continuous-time nonlinear control systems

The paper develops the algebraic formalism that is based on the language of ideals and modules associated with the analytic control system given by the ordinary differential equations. Using the Nagano theorem we show how the integral manifolds of Lie algebra of the considered system can be determined by the generators of some ideal of the ring of germs of analytic functions that is invariant with respect to germs of vector fields from the Lie algebra associated with the system. Additionally, basing on the language of ideals and modules the strong accessibility problem is studied. The singular points where the rank of (co)distributions associated to the system is different than at points from their neighbourhoods are considered. Using the germs of one-forms associated with the generators of the ideal one can define the integrable codistribution that in general is not analytic.

Local topological algebraicity with algebraic coefficients of analytic sets or functions

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.

Milnor and Tjurina numbers for a hypersurface germ with isolated singularity

Assume that f:(Cn,0)→(C,0) is an analytic function germ at the origin with only isolated singularity. Let μ and τ be the corresponding Milnor and Tjurina numbers. We show that μτ≤n. As an application, we give a lower bound for the Tjurina number in terms of n and the multiplicity of f at the origin.

Blow-analytic equivalence versus contact bi-Lipschitz equivalence

The main result of this note is that two blow-analytically equivalent real analytic plane function germs are sub-analytically bi-Lipschitz contact equivalent.

On finite determinacy for matrices of power series

Let $$R=K[[x_1,\ldots ,x_s]]$$ 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 equivalence relations given by the action on $$M_{m,n}$$ of a group G. G can be the group of automorphisms of R, combined with the multiplication of invertible matrices from the left, from the right, or from both sides, respectively. We call A finitely G-determined if A is G-equivalent to any matrix B with $${A-B} \in {\mathfrak {m}^k M_{m,n}}$$ for some finite integer k, which implies in particular that A is G-equivalent to a matrix with polynomial entries. The classical criterion for analytic or differential map germs $$f:(K^s,0) \rightarrow (K^m,0)$$ , $$K = \mathbb {R}, \mathbb {C}$$ , says that $$f \in M_{m,1}$$ is finitely determined (with respect to various group actions) iff the tangent space to the orbit of f has finite codimension in $$M_{m,1}$$ . We extend this criterion to arbitrary matrices in $$M_{m,n}$$ if the characteristic of K is 0 or, more general, if the orbit map is separable. In positive characteristic however, the problem is more subtle since the orbit map is in general not separable, as we show by an example. This fact had been overlooked in previous papers. Our main result is a general sufficient criterion for finite G-determinacy in $$M_{m,n}$$ in arbitrary characteristic in terms of the tangent image of the orbit map, which we introduce in this paper. This criterion provides a computable bound for the G-determinacy of a matrix A in $$M_{m,n}$$ , which is new even in characteristic 0.

Integral closure and bounds for quotients of multiplicities of monomial ideals

Given a pair of monomial ideals I and J of finite colength of the ring of analytic function germs (Cn,0)→C, we prove that some power of I admits a reduction formed by homogeneous polynomials with respect to the Newton filtration induced by J if and only if the quotient of multiplicities e(I)/e(J) attains a suitable upper bound expressed in terms of the Newton polyhedra of I and J. We also explore other connections between mixed multiplicities, Newton filtrations and the integral closure of ideals.

The Special Closure of Polynomial Maps and Global Non-degeneracy

Let \(F:\mathbb C^n\rightarrow \mathbb C^n\) be a polynomial map such that \(F^{-1}(0)\) is finite. We analyze the connections between the multiplicity of F, the Newton polyhedron of F and the set of special monomials with respect to F, which is a notion motivated by the integral closure of ideals in the ring of analytic function germs \((\mathbb C^n,0)\rightarrow \mathbb C\). In particular, we characterize the polynomial maps whose set of special monomials is maximal.

Arc-wise analytic stratification, Whitney fibering conjecture and Zariski equisingularity

In this paper we show Whitney's fibering conjecture in the real and complex, local analytic and global algebraic cases.For a given germ of complex or real analytic set, we show the existence of a stratification satisfying a strong (real arc-analytic with respect to all variables and analytic with respect to the parameter space) trivialization property along each stratum. We call such a trivialization arc-wise analytic and we show that it can be constructed under the classical Zariski algebro-geometric equisingularity assumptions. Using a slightly stronger version of the Zariski equisingularity, we show the existence of Whitney's stratified fibration, satisfying the conditions (b) of Whitney and (w) of Verdier. Our construction is based on the Puiseux with parameter theorem and a generalization of Whitney's interpolation. For algebraic sets our construction gives a global stratification.We also present several applications of the arc-wise analytic trivialization, mainly to the stratification theory and the equisingularity of analytic set and function germs. In the real algebraic case, for an algebraic family of projective varieties, we show that the Zariski equisingularity implies local constancy of the associated weight filtration.

Local observability of nonlinear positive continuous-time systems

Two observability properties of continuous-time nonlinear positive systems are studied: local observability and local positive observability. An algebraic necessary and sufficient condition for local observability is proved. It is expressed with the aid of ideals and cones in the ring of germs of analytic functions. Local positive observability is characterized in a more geometric fashion. This characterization naturally extends the known characterization of positive observability of positive linear systems. As for linear systems, criteria for local positive observability for nonlinear systems are very restrictive. This result is parallel to characterization of local positive reachability of nonlinear systems.

Local topological algebraicity of analytic function germs

Mostowski showed that every (real or complex) germ of an analytic set is homeomorphic to the germ of an algebraic set. In this paper we show that every (real or complex) analytic function germ, defined on a possibly singular analytic space, is topologically equivalent to a polynomial function germ defined on an affine algebraic variety.

The Euler obstruction of a function on a determinantal variety and on a curve

Given an analytic function germ f: (X, 0) → ℂ on an isolated determinantal singularity or on a reduced curve, we present formulas relating the local Euler obstruction of f to the vanishing Euler characteristic of the fiber X ∩ f-1(0) and to the Milnor number of f. Restricting ourselves to the case where X is a complete intersection, we obtain an easy way to calculate the local Euler obstruction of f as the difference between the dimension of two algebras.

Moduli spaces for topologically quasi-homogeneous functions

We consider the topological class of a germ of 2-variables quasi-homogeneous complex analytic function. Each element f in this class induces a germ of foliation (f = constants) and a germ of curve (f = 0). We first describe the moduli space of the foliations in this class and give analytic normal forms. The classification of curves induces a distribution on this moduli space. By studying the infinitesimal generators of this distribution, we can compute the generic dimension of the moduli space for the curves, and we obtain the corresponding generic normal forms.

Gauss words and the topology of map germs from $\mathbb R^3$ to $\mathbb R^3$

The link of a real analytic map germ f\colon (\mathbb{R}^{3}, 0) \to (\mathbb{R}^{3}, 0) is obtained by taking the intersection of the image with a small enough sphere S^2_\epsilon centered at the origin in \mathbb R^3 . If f is finitely determined, then the link is a stable map \gamma from S^2 to S^2 . We define Gauss words which contains all the topological information of the link in the case that the singular set S(\gamma) is connected and we prove that in this case they provide us with a complete topological invariant.

A’Campo curvature bumps and the Dirac phenomenon near a singular point

The level curves of an analytic function germ can have bumps (maxima of Gaussian curvature) at unexpected points near the singularity. This phenomenon is fully explored for f ( z , w ) ∈ C { z , w } , using the Newton–Puiseux infinitesimals and the notion of gradient canyon. Equally unexpected is the Dirac phenomenon: as c → 0 , the total Gaussian curvature of f = c accumulates in the minimal gradient canyons, and nowhere else. Our approach mimics the introduction of polar coordinates in Analytic Geometry.

On the bi-Lipschitz contact equivalence of plane complex function-germs

In this short note, we consider the problem of bi-Lipschitz contact equivalence of complex analytic function-germs of two variables. It is inquiring about the infinitesimal sizes of such function-germs, up to bi-Lipschitz changes of coordinates. We show that this problem is equivalent to the problem of the right topological classification.

A Note on the Weierstrass Preparation Theorem in Quasianalytic Local Rings

AbstractConsider quasianalytic local rings of germs of smooth functions closed under composition, implicit equation, and monomial division. We show that if theWeierstrass Preparation Theoremholds in such a ring, then all elements of it are germs of analytic functions.