Germs Of Analytic Functions Research Articles

Monodromy zeta-functions of deformations and Newton diagrams

For a one-parameter deformation of an analytic complex function germ of several variables, there is defined its monodromy zeta-function. We give a Varchenko type formula for this zeta-function if the deformation is non-degenerate with respect to its Newton diagram.

Refinements of Milnor's fibration theorem for complex singularities

Let X be an analytic subset of an open neighbourhood U of the origin 0 ̲ in C n . Let f : ( X , 0 ̲ ) → ( C , 0 ) be holomorphic and set V = f − 1 ( 0 ) . Let B ε be a ball in U of sufficiently small radius ε > 0 , centred at 0 ̲ ∈ C n . We show that f has an associated canonical pencil of real analytic hypersurfaces X θ , with axis V, which leads to a fibration Φ of the whole space ( X ∩ B ε ) ∖ V over S 1 . Its restriction to ( X ∩ S ε ) ∖ V is the usual Milnor fibration ϕ = f | f | , while its restriction to the Milnor tube f − 1 ( ∂ D η ) ∩ B ε is the Milnor–Lê fibration of f. Each element of the pencil X θ meets transversally the boundary sphere S ε = ∂ B ε , and the intersection is the union of the link of f and two homeomorphic fibres of ϕ over antipodal points in the circle. Furthermore, the space X ˜ obtained by the real blow up of the ideal ( Re ( f ) , Im ( f ) ) is a fibre bundle over RP 1 with the X θ as fibres. These constructions work also, to some extent, for real analytic map-germs, and give us a clear picture of the differences, concerning Milnor fibrations, between real and complex analytic singularities.

The directional dimension of subanalytic sets is invariant under bi-Lipschitz homeomorphisms

Let A ⊂ R n be a set-germ at 0 ∈ R n such that 0 ∈ A. We say that r ∈ S n−1 is a direction of A at 0 ∈ R n if there is a sequence of points {xi} ⊂ A\{0} tending to 0 ∈ R n such that x i k xik → r as i → ∞. Let D(A) denote the set of all directions of A at 0 ∈ R n . Let A, B ⊂ R n be subanalytic set-germs at 0 ∈ R n such that 0 ∈ A ∩ B. We study the problem of whether the dimension of the common direction set, dim(D(A) ∩ D(B)) is preserved by bi-Lipschitz homeomorphisms. We show that although it is not true in general, it is preserved if the images of A and B are also subanalytic. In particular if two subanalytic set-germs are bi-Lipschitz equivalent their direction sets must have the same dimension. Let us recall the notion of blow-analyticity. Let f, g : (R n ,0) → (R,0) be analytic function-germs. We say that they are blow-analytically equivalent if there are real modifications � : (M, � −1 (0)) → (R n ,0), � ' : (M ' , � '−1 (0)) → (R n ,0) and an analytic isomorphism � : (M, � −1 (0)) → (M ' , � '−1 (0)) which induces a homeomorphism φ : (R n ,0) → (R n ,0), � ' ◦ � = φ ◦ � , such that f = g ◦ φ. A blow-analytic homeomorphism is such a φ, a homeomorphism induced by an analytic isomorphism via real modifications.

A criterion for topological equivalence of two variable complex analytic function germs

We show that two analytic function germs $(\mathbf{C}^{2},0) \to (\mathbf{C},0)$ are topologically right equivalent if and only if there is a one-to-one correspondence between the irreducible components of their zero sets that preserves the multiplicites of these components, their Puiseux pairs, and the intersection numbers of any pair of distinct components.

$C^{\ell}$-$G$-triviality of map germs and Newton polyhedra, $G = \mathcal R$, $\mathcal C$ and $\mathcal K$

We provide estimates for the $C^{\ell}$-$G$-triviality, for $ 0 \leq \ell < \infty$ and $G$ is one of Mather's groups ${\mathcal R}$, ${\mathcal C}$ or ${\mathcal K}$, of deformations of analytic map germs $f: (\mathbb{R}^n,0) \to (\mathbb{R}^p,0)$ of type $f_t(x)=f(x)+θ(x,t)$ which satisfy a non-degeneracy condition with respect to some Newton polyhedron. We apply the method of construction of controlled vector fields and, for each group $G$, the control function is determined from the choice of a convenient {\it Newton filtration } in the ring of real analytic germs. The results are given in terms of the filtration of the coordinate function germs $f_1, \ldots , f_p$ of $f$.

On the positive extension property and Hilbert's 17th problem for real analytic sets

In this work we study the Positive Extension property and Hilbert's 17th problem for real analytic germs and sets. A real analytic germ X0 of has the property if every positive semidefinite analytic function germ on X0 has a positive semidefinite analytic extension to ; analogously one states the property for a global real analytic set X in an open set Ω of ℝn. These properties are natural variations of Hilbert's 17th problem. Here, we prove that: (1) A real analytic germ has the property if and only if every positive semidefinite analytic function germ on X0 is a sum of squares of analytic function germs on X0; and (2) a global real analytic set X of dimension ≦ 2 and local embedding dimension ≦ 3 has the property if and only if it is coherent and all its germs have the property. If that is the case, every positive semidefinite analytic function on X is a sum of squares of analytic functions on X. Moreover, we classify the singularities with the property.

Un résultat de connexité pour les variétés analytiques p-adiques: privilège et noethérianité

AbstractLet k be a non-Archimedean field, let X be a k-affinoid space and let f1,…,fn, with $n\in \mathbb {N}^*$, be analytic functions over X. If X is irreducible, we prove that the analytic domain $\bigcup _{1\le j\le n} \{x\in X\mid |f_{j}(x)|\ge \varepsilon _{j}\}$ is still irreducible, provided that $(\varepsilon _{1},\ldots ,\varepsilon _{n}) \in \mathbb {R}_{+}^n$ is small enough. Then, for a general X, we precisely describe how the geometric connected components of the spaces $\{x\in X\mid |f(x)|\ge \varepsilon \}$ behave with regards to ε. Finally, we obtain a result concerning privileged neighbourhoods and adapt a theorem from complex analytic geometry about Noetherianity for germs of analytic functions.

On The Effective Computation Of Łojasiewicz Exponents Via Newton Polyhedra

In this paper we give some conclusions on Newton non-degenerate analytic map germs on Kn (K = ℝ or ℂ), using information from their Newton polyhedra. As a consequence, we obtain the exact value of the Lojasiewicz exponent at the origin of Newton non-degenerate analytic map germs. In particular, we establish a connection between Newton non-degenerate ideals and their integral closures, thus leading to a simple proof of a result of Saia. Similar results are also considered to polynomial maps which are Newton non-degenerate at infinity.

C 0-Sufficiency, Kuiper–Kuo and Thom Conditions for Non-isolated Singularity

In this paper, a criterion on the C0-sufficiency for a function germ with non-isolated singularity is obtained analogously to that of Kuiper–Kuo for the case of isolated singularities. Moreover, the Kuiper–Kuo condition and the Thom condition for an analytic function germ with non-isolated singularity are proved to be equivalent.

Noethérianité et privilège en géométrique analytique p-adique

Let k be a non-Archimedean field, X a k-affinoid space and f an analytic function over X. We describe precisely how the geometric connected components of the spaces { x ∈ X | | f ( x ) | ⩾ ε } behave with regards to ε. We also obtain a result concerning privileged neighbourhoods and adapt a theorem from complex analytic geometry about Noetherianity for germs of analytic functions. To cite this article: J. Poineau, C. R. Acad. Sci. Paris, Ser. I 343 (2006).

On the Multiplicity of a C ∞–differentiable Function–germ

Risler & Trotman in 1997 proved that the multiplicity of an analytic function germ is left–right lipschitz invariant, which provided a partial answer to Zariski conjecture. In this note, based on the recent work of Comte, Milman & Trotman, we generalize the work of them to prove that the multiplicity of a C∞–differentiable function germ is also left–right lipschitz invariant.

Duality of the space of germs of harmonic vector fields on a compact

A vector field u ¯ in R 3 is said to be harmonic in the open set U if rot u ¯ = 0 ¯ , div u ¯ = 0 in U. Harmonic vector fields are a natural extension to R 3 of the concept of analytic function of complex variable. We characterize continuous linear functionals acting on the space h ( K ) of germs of harmonic vector fields on a compact set K. This result provides an R 3 -analog of a theorem by G. Köthe on the dual of the space of germs of analytic functions of complex variable on a compact. To cite this article: R. Dáger, A. Presa, C. R. Acad. Sci. Paris, Ser. I 343 (2006).

The corank and the index are blow-Nash invariants

Is the corank an invariant of the blow-analytic equivalence between real analytic function germs? We give a positive answer in the case of the blow-Nash equivalence, which is a natural variant of the blow-analytic equivalence for Nash function germs. In addition, we prove that the index is also a blow-Nash invariant. Their proof is based on the computation of some virtual Poincaré polynomials and zeta functions associated to a Nash function germ.

Topological triviality of families of real isolated singularities and their Milnor fibrations

The aim of this paper is to study the topological triviality and the topological equivalence of the Milnor fibrations for families of real analytic map germs with no coalescing of critical points.

JACOBIAN IDEALS AND THE NEWTON NON-DEGENERACY CONDITION

AbstractIn this paper we extract some conclusions about Newton non-degenerate ideals and the computation of Łojasiewicz exponents relative to this kind of ideal. This motivates us to study the Newton non-degeneracy condition on the Jacobian ideal of a given analytic function germ $f:(\mathbb{C}^n,0)\to(\mathbb{C},0)$. In particular, we establish a connection between Newton non-degenerate functions and functions whose Jacobian ideal is Newton non-degenerate.AMS 2000 Mathematics subject classification: Primary 32S05. Secondary 57R45

On the Pythagoras numbers of real analytic set germs

We Show that (i) the Pythagoras number of a real analytic set germ is the supremum of the Pythagoras numbers of the curve germs it contains, and (ii) every real analytic curve germ is contained in a real analytic surface germ with the same Pythagoras number (or Pythagoras number 2 if the curve is Pythagorean). This gives new examples and counterexamples concerning sums of squares and positive semidefinite analytic function germs.

Topological finite-determinacy of functions with non-isolated singularities

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 following statement which generalizes classical results of Thom and Varchenko: let A be the complement in the ideal I of the space of germs whose topological type remains unchanged under a deformation within the ideal that only modifies sufficiently large order terms of the Taylor expansion. Then A has infinite codimension in I in a suitable sense. We also prove the existence of generic topological types of families of germs of I parametrized by an irreducible analytic set.

An approximation theorem for entire functions of exponential type and stability of zero sequences

Let be an entire function of exponential type in with indicator function ; let , , be a subsequence of zeros of the entire function of exponential type ; let be a complex number sequence and assume that A simple construction of a sequence of entire functions of exponential type transforming into a subsequence of zeros of an entire function of exponential type such that is put forward (an approximation theorem). This result is applied to stability problems of zero sequences and non-uniqueness sequences for spaces of entire functions of exponential type with constraints on the indicators and to the problem of the stability of the completeness property of exponential systems in the space of germs of analytic functions on a compact convex set.

Algebraic and analytic finite spaces of orderings

In this paper, the finite subspaces of orderings of the ring of regular functions on an algebraic set V are compared with those of the ring of analytic function germs at a point of V. Necessary and sufficient conditions for subspaces to be isomorphic are given, both from a purely algebraic and from a more geometric point of view. As a result, a criterion for analytic separation of semialgebraic sets is proved. Extendability of such subspaces is also proved to be stable under suitable approximations.

Analytic surface germs with minimal Pythagoras number

We determine all complete intersection surface germs whose Pythagoras number is 2, and find that they are all embedded in ℝ3 and have the property that every positive semidefinite analytic function germ is a sum of squares of analytic function germs. In addition, we discuss completely these properties for mixed surface germs in ℝ3. Finally, we find in higher embedding dimension three different families with these same properties.