Germs Of Functions Research Articles

Random local complex dynamics

The study of the dynamics of an holomorphic map near a fixed point is a central topic in complex dynamical systems. In this paper, we will consider the corresponding random setting: given a probability measure $\unicode[STIX]{x1D708}$ with compact support on the space of germs of holomorphic maps fixing the origin, we study the compositions $f_{n}\circ \cdots \circ f_{1}$, where each $f_{i}$ is chosen independently with probability $\unicode[STIX]{x1D708}$. As in the deterministic case, the stability of the family of the random iterates is mostly determined by the linear part of the germs in the support of the measure. A particularly interesting case occurs when all Lyapunov exponents vanish, in which case stability implies simultaneous linearizability of all germs in $\text{supp}(\unicode[STIX]{x1D708})$.

Transseries as germs of surreal functions

We show that Écalle’s transseries and their variants (LE and EL-series) can be interpreted as functions from positive infinite surreal numbers to surreal numbers. The same holds for a much larger class of formal series, here called omega-series. Omega-series are the smallest subfield of the surreal numbers containing the reals, the ordinal omega, and closed under the exp and log functions and all possible infinite sums. They form a proper class, can be composed and differentiated, and are surreal analytic. The surreal numbers themselves can be interpreted as a large field of transseries containing the omega-series, but, unlike omega-series, they lack a composition operator compatible with the derivation introduced by the authors in an earlier paper.

Recognition of unimodal map germs from the plane to the plane by invariants

In this paper, we characterize the classification of unimodal maps from the plane to the plane with respect to [Formula: see text]-equivalence given by Rieger in terms of invariants. We recall the classification over an algebraically closed field of characteristic [Formula: see text]. On the basis of this characterization, we present an algorithm to compute the type of the unimodal maps from the plane to the plane without computing the normal form and also give its implementation in the computer algebra system Singular.

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.

Integration over the Space of Functions and Poincaré Series Revisited

Earlier (2000) the authors introduced the notion of the integral with respect to the Euler characteristic over the space of germs of functions on a variety and over its projectivization. This notion allowed the authors to rewrite known definitions and statements in new terms and also turned out to be an effective tool for computing the Poincar´e series of multi-index filtrations in some situations. However, the “classical” (initial) notion can be applied only to multi-index filtrations defined by so-called finitely determined valuations (or order functions). Here we introduce a modified version of the notion of the integral with respect to the Euler characteristic over the projectivization of the space of function germs. This version can be applied in a number of settings where the “classical approach” does not work. We give examples of the application of this concept to definitions and computations of the Poincar´e series (including equivariant ones) of collections of plane valuations which contain valuations not centred at the origin.

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.

Finiteness theorem for multi‐‐bi‐Lipschitz equivalence of map germs

AbstractLet be the set of all real polynomial map germs with degree of less than or equal to . The main result of this paper shows that the set of equivalence classes of , with respect to multi‐‐bi‐Lipschitz equivalence, is finite.

One-Side Continuity of Meromorphic Mappings Between Real Analytic Hypersurfaces

We prove that a meromorphic mapping, which sends a peace of a real analytic strictly pseudoconvex hypersurface in $${\mathbb C}^2$$ to a compact subset of $${\mathbb C}^N$$ which does not contain germs of non-constant complex curves is continuous from the concave side of the hypersurface. This implies the analytic continuability along CR-paths of germs of holomorphic mappings from real analytic hypersurfaces with non-vanishing Levi form to the locally spherical ones in all dimensions.

A Remark on Two Notions of Order of Contact

We recall two measurements of the order of contact of an ideal in the ring of germs of holomorphic functions at a point, and we provide a class of examples in which they differ.

Global bifurcations in the two-sphere: a new perspective

We construct an open set of structurally unstable three parameter families whose weak and so called moderate topological classification defined below has a numerical invariant that may take an arbitrary positive value. Here and below "families" are "families of vector fields in the two-sphere". This result disproves an Arnold's conjecture of 1985. Then we construct an open set of six parameter families whose moderate topological classification has a functional invariant. This invariant is an arbitrary germ of a smooth map $(\mathbb R_+, a)\to(\mathbb R_+, b)$. More generally, for any positive integers $d$ and $d'$, we construct an open set of families whose topological classification has a germ of a smooth map $\left(\mathbb R_+^d, a\right)\to\left(\mathbb R_+^{d'}, b\right)$ as an invariant. Any smooth germ of this kind may be realized as such an invariant. These results open a new perspective of the global bifurcation theory in the two sphere. This perspective is discussed at the end of the paper.

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.

On the arc-analytic type of some weighted homogeneous polynomials

It is known that the weights of a complex weighted homogeneous polynomial $f$ with isolated singularity are analytic invariants of $(\mathbb C^d,f^{-1}(0))$. When $d=2,3$ this result holds by assuming merely the topological type instead of the analytic one. G. Fichou and T. Fukui recently proved the following real counterpart: the blow-Nash type of a real singular non-degenerate convenient weighted homogeneous polynomial in three variables determines its weights. The aim of this paper is to generalize the above-cited result with no condition on the number of variables. We work with a characterization of the blow-Nash equivalence called the arc-analytic equivalence. It is an equivalence relation on Nash function germs with no continuous moduli which may be seen as a semialgebraic version of the blow-analytic equivalence of T.-C. Kuo.

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.

Whitney equisingularity of families of surfaces in ℂ3

AbstractIn this paper, we study families of singular surfaces in ℂ3 parametrised by $\mathcal {A}$-finitely determined map germs. We consider the topological triviality and Whitney equisingularity of an unfolding F of a finitely determined map germ f : (ℂ2, 0) → (ℂ3, 0). We investigate the following question: topological triviality implies Whitney equisingularity of the unfolding F? We provide a complete answer to this question, by giving counterexamples showing how the conjecture can be false.

Log-canonical thresholds in real and complex dimension 2

We study the set of log-canonical thresholds (or critical integrability indices) of holomorphic (resp. real analytic) function germs in ℂ 2 (resp. ℝ 2 ). In particular, we prove that the ascending chain condition holds, and that the positive accumulation points of decreasing sequences are precisely the integrability indices of holomorphic (resp. real analytic) functions in dimension 1. This gives a new proof of a theorem of Phong–Sturm.

Thom irregularity and Milnor tube fibrations

We find natural and convenient conditions which allow us to produce classes of genuine real map germs with Milnor tube fibration, either with Thom regularity or without it.

Periodic complex map germs and foliations.

In this paper we study topological and analytical conditions on the orbits of a germ of diffeomorphism in the complex plane in order to obtain periodicity. In particular, we give a simple proof of a finiteness criteria for groups of analytic diffeomorphisms, stated in Brochero Martínez 2003. As an application, we derive some consequences about the integrability of complex vector fields in dimension three in a neighborhood of a singular point.

An algorithm to compute a presentation of pushforward modules

We describe an algorithm to compute a presentation of the pushforward module f⁎OX for a finite map germ f:X→(Cn+1,0), where X is Cohen–Macaulay of dimension n. The algorithm is an improvement of a method by Mond and Pellikaan. We give applications to problems in singularity theory, computed by means of an implementation in the software Singular.