Germs Of Analytic Functions Research Articles

Topology of Fold Map Germs from mathbb R^3 to mathbb R^5

Let f:(mathbb R^3,0)rightarrow (mathbb R^5,0) be an analytic map germ with isolated instability. Its link is a stable map which is obtained by taking the intersection of the image of f with a small enough sphere S^4_epsilon centered at the origin in mathbb R^5. If f is of fold type, we define a labeled tree associated to its link and prove that is a complete topological invariant for it. As an application we obtain the complete topological classification of map germs contained in the mathcal {A}^2-class (x,y,z^2,xz,0).

On the Topology of the Milnor Fibration

In this paper we present new results about the topology of the Milnor fibrations of analytic function-germs with a special attention to the topology of the fibers. In particular, we provide a short review on the existence of the Milnor fibrations in the real and complex cases. This allows us to compare our results with the previous ones.

Thom condition and monodromy

We give the definition of the Thom condition and we show that given any germ of complex analytic function f:(X,x)rightarrow ({mathbb {C}},0) on a complex analytic space X, there exists a geometric local monodromy without fixed points, provided that fin {mathfrak {m}}_{X,x}^2, where {mathfrak {m}}_{X,x} is the maximal ideal of {mathcal {O}}_{X,x}. This result generalizes a well-known theorem of the second named author when X is smooth and proves a statement by Tibar in his PhD thesis. It also implies the A’Campo theorem that the Lefschetz number of the monodromy is equal to zero. Moreover, we give an application to the case that X has maximal rectified homotopical depth at x and show that a family of such functions with isolated critical points and constant total Milnor number has no coalescing of singularities.

The dual tree of a fold map germ from to

Let $f\colon (\mathbb {R}^{3},0)\to (\mathbb {R}^{4},0)$ be an analytic map germ with isolated instability. Its link is a stable map which is obtained by taking the intersection of the image of $f$ with a small enough sphere $S^{3}_\epsilon$ centred at the origin in $\mathbb {R}^{4}$. If $f$ is of fold type, we define a tree, that we call dual tree, that contains all the topological information of the link and we prove that in this case it is a complete topological invariant. As an application we give a procedure to obtain normal forms for any topological class of fold type.

Pluriharmonic Definable Functions in Some O-Minimal Expansions of the Real Field

In this paper, we first try to solve the following problem: If a pluriharmonic function $f$ is definable in an arbitrary o-minimal expansion of the structure of the real field $\overline{\mathbb{R}}:=(\mathbb{R},+,-,.,0,1,<)$, does this function be locally the real part of a holomorphic function which is definable in the same expansion? In Proposition 2.1 below, we prove that this problem has a positive answer if the Weierstrass division theorem holds true for the system of the rings of real analytic definable germs at the origin of $\mathbb{R}^n$. We obtain the same answer for an o-minimal expansion of the real field which is pfaffian closed (Proposition 2.6) for the harmonic functions. In the last section, we are going to show that the Weierstrass division theorem does not hold true for the rings of germs of real analytic functions at $0\in\mathbb{R}^n$ which are definable in the o-minimal structure $(\overline{\mathbb{R}}, x^{\alpha_1},\ldots,x^{\alpha_p})$, here $\alpha_1,\ldots,\alpha_p$ are irrational real numbers.

The polynomial Hermite-Padé -system for meromorphic functions on a compact Riemann surface

Given a tuple of germs of arbitrary analytic functions at a fixed point, we introduce the polynomial Hermite-Padé -system, which includes the Hermite-Padé polynomials of types I and II. In the generic case we find the weak asymptotics of the polynomials of the Hermite-Padé -system constructed from the tuple of germs of functions that are meromorphic on an -sheeted compact Riemann surface . We show that if for some meromorphic function on , then with the help of the ratios of polynomials of the Hermite-Padé -system we recover the values of on all sheets of the Nuttall partition of , apart from the last sheet. Bibliography: 18 titles.

A proof of A. Gabrielov’s rank theorem

This article contains a complete proof of Gabrielov’s rank theorem, a fundamental result in the study of analytic map germs. Inspired by the works of Gabrielov and Tougeron, we develop formal-geometric techniques which clarify the difficult parts of the original proof. These techniques are of independent interest, and we illustrate this by adding a new (very short) proof of the Abhyankar-Jung theorem. We include, furthermore, new extensions of the rank theorem (concerning the Zariski main theorem and elimination theory) to commutative algebra.

Uniform stable radius and Milnor number for non-degenerate isolated complete intersection singularities

We prove that for two germs of analytic mappings $$f,g:({\mathbb {C}}^n,0) \rightarrow ({\mathbb {C}}^p,0)$$ with the same Newton polyhedra which are (Khovanskii) non-degenerate and their zero sets are complete intersections with isolated singularity at the origin, there is a piecewise analytic family $$\{f_t\}$$ of analytic maps with $$f_0=f, f_1=g$$ which has a so-called uniform stable radius for the Milnor fibration. As a corollary, we show that their Milnor numbers are equal. Also, a formula for the Milnor number is given in terms of the Newton polyhedra of the component functions. This is a generalization of the result by C. Bivia-Ausina. Consequently, we obtain that the Milnor number of a non-degenerate isolated complete intersection singularity is an invariant of Newton boundaries.

Holonomy and Equivalence of Analytic Foliations

The main goal of this paper is the analytic classification of the germs of singular foliations generated, up to an analytic change of coordinates, by the germs of vector fields of form the \(x\partial _{x}+{\sum }_{i=1}^{n}a_{i}(x,\mathbf {z})\partial _{z_{i}}\), where ai(x,z) is a germ of analytic function with ai(x,0) = 0. We prove, under some hypothesis, that these germs of singular foliations are analytically classified once their local holonomy along a given separatrix is analytically conjugated.

Topological Classification and Finite Determinacy of Knotted Maps

We show that the knot type of the link of a real analytic map germ with isolated singularity f : ( R 2 , 0 ) → ( R 4 , 0 ) is a complete invariant for C 0 - A -equivalence. Moreover, we also prove that isolated singularity implies finite C 0 -determinacy, giving an explicit estimate for its degree. For the general case of real analytic map germs, f : ( R n , 0 ) → ( R p , 0 ) ( n ≤ p ), we use the Lojasiewicz exponent associated with Mond’s double point ideal I 2 ( f ) to obtain some criteria of Lipschitz and analytic regularity.

Analytic solutions of convolution equations on convex sets in the complex plane with an open obstacle on the boundary

Conditions, including criteria, are established for the existence of a continuous linear right inverse to a surjective convolution operator in the space of germs of analytic functions on a convex subset of the complex plane which has a countable neighbourhood basis consisting of convex domains. These are stated in terms of the existence of special families of subharmonic functions and the boundary behaviour of convex conformal mappings related to the sets in question. Bibliography: 50 titles.

Finite $C^0$-determinacy of real analytic map germs with isolated instability

Let f : (Rn; 0) (Rp; 0) be a real analytic map germ with isolated instability. We prove that if n = 2 and p = 2; 3, then f is finitely C0-determined. This result can be seen as a weaker real counterpart of Mather-Gaffney finite determinacy criterion.

The dimension of the moduli spaces of curves defined by 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. The 0-level of such a function defines a germ of analytic curve. Proceeding similarly to the homogeneous case Genzmer and Paul (2011) and the quasi homogeneous case Genzmer and Paul (2016), we describe an algorithm which computes the dimension of the generic strata of the local moduli space of curves.

Tauberian theorems for k–summability with respect to an analytic germ

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 variable and in monomials, and a method of monomialization of germs of analytic functions.

Mixed Bruce–Roberts numbers

AbstractWe extend the notions of μ*-sequences and Tjurina numbers of functions to the framework of Bruce–Roberts numbers, that is, to pairs formed by the germ at 0 of a complex analytic variety X ⊆ ℂn and a finitely ${\mathcal R}(X)$-determined analytic function germ f : (ℂn, 0) → (ℂ, 0). We analyze some fundamental properties of these numbers.

Double points in families of map germs from ℝ2 to ℝ3

We show that a 1-parameter family of real analytic map germs [Formula: see text] with isolated instability is topologically trivial if it is excellent and the family of double point curves [Formula: see text] in [Formula: see text] is topologically trivial. In particular, we deduce that [Formula: see text] is topologically trivial when the Milnor number [Formula: see text] is constant.

The sequence of mixed Łojasiewicz exponents associated to pairs of monomial ideals

We analyze the sequence LJ⁎(I) of mixed Łojasiewicz exponents attached to any pair I,J of monomial ideals of finite colength of the ring of analytic function germs (Cn,0)→C. In particular, we obtain a combinatorial expression for this sequence when J is diagonal. We also show several relations of LJ⁎(I) with other numerical invariants associated to I and J.

On the growth behaviour of Hironaka quotients

We consider a finite analytic morphism φ = (f, g) : (X, p) −→ (C 2 , 0) where (X, p) is a complex analytic normal surface germ and f and g are complex analytic function germs. Let π : (Y, E Y) → (X, p) be a good resolution of φ with exceptional divisor E Y = π −1 (p). We denote G(Y) the dual graph of the resolution π. We study the behaviour of the Hironaka quotients of (f, g) associated to the vertices of G(Y). We show that there exists maximal oriented arcs in G(Y) along which the Hironaka quotients of (f, g) strictly increase and they are constant on the connected components of the closure of the complement of the union of the maximal oriented arcs.

Images of analytic map germs and singular fibrations

For a map germ $G$ with target $(\mathbb C^{p}, 0)$ or $(\mathbb R^{p}, 0)$ with $p\ge 2$, we address two phenomena which do not occur when $p=1$: the image of $G$ may be not well-defined as a set germ, and a local fibration near the origin may not exist. We show how these two phenomena are related, and how they can be characterised.

On Tangent Cones of Analytic Sets and $$\L $$ojasiewicz Exponents

For an analytic set V in $${\mathbb {K}}^n,$$$$\mathbb K={\mathbb {R}}$$ or $${\mathbb {C}}$$, which contains the origin $$0\in \mathbb K^n$$, the geometric tangent cone of V at 0 is the set of vectors in $${\mathbb {K}}^n$$ which are the limits of secant lines passing through the origin and non-zero sequences in V that converge to the origin; the algebraic tangent cone of V at 0 is the algebraic set defined by the ideal generated by the initial forms of all analytic functions f whose germs at 0 are in the ring of germs of analytic functions in $${\mathbb {K}}^{n}$$ about 0 which vanish on the germ of V at 0. In this paper, we give some characterization for the geometric tangent cone and compare these two tangent cones of V at 0. In particular, we characterize the geometric tangent cone of V at 0 via the so-called $$\L $$ojasiewicz exponent of an analytic map germ along a line and compute this number in some special cases.