Normal forms for singularities of pedal curves produced by non-singular dual curve germs in <Emphasis Type="Italic">S</Emphasis><Superscript><Emphasis Type="Italic">n</Emphasis></Superscript>

For an n-dimensional spherical unit speed curve r and a given point P, we can define naturally the pedal curve of r relative to the pedal point P. When the dual curve germs are non-singular, singularity types of such pedal curves depend only on locations of pedal points. In this paper, we give a complete list of normal forms for singularities and locations of pedal points when the dual curve germs are non-singular. As an application of our list, we characterize C ^∞ left equivalence classes of pedal curve germs (I, s ₀) → S ⁿ produced by non-singular dual curve germ from the viewpoint of the relation between tangent space and tangent space.      

Holomorphic Linearization of Commuting Germs of Holomorphic Maps

Let f ₁,…,f _h be h≥2 germs of biholomorphisms of ? ⁿ fixing the origin. We investigate the shape that a (formal) simultaneous linearization of the given germs can have, and we prove that if f ₁,…,f _h commute and their linear parts are almost simultaneously Jordanizable, then they are simultaneously formally linearizable. We next introduce a simultaneous Brjuno-type condition and prove that, in case the linear terms of the germs are diagonalizable, if the germs commute and our Brjuno-type condition holds, then they are holomorphically simultaneously linearizable. This answers a multi-dimensional version of a problem raised by Moser.     

Criteria for the injectivity of analytic sheaves

It is shown that a moduleL over the sheafO of germs of holomorphic functions on a domain G of Cⁿ is injective if and only if the following conditions are satisfied; a)L is flabby; b) for every closed set S ?G and every point z λ G, the stalk se _z of the sheafS ^L;U1→Γ S ^(U:L) is an injectiveO _z -module. It follows in particular that the sheaf of germs of hyperfunctions is injective over the sheaf of germs of analytic functions.     

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.     

A family of non-isomorphism results

We calculate the local groups of germs associated with the higher dimensional R. Thompson groups nV. For a given \({n\in N\cup\left\{\omega\right\}}\) , these groups of germs are free abelian groups of rank r, for r ≤ n (there are some groups of germs associated with nV with rank precisely k for each index 1 ≤ k ≤ n). By Rubin’s theorem, any conjectured isomorphism between higher dimensional R. Thompson groups induces an isomorphism between associated groups of germs. Thus, if m ≠ n the groups mV and nV cannot be isomorphic. This answers a question of Brin.     

Dynamics of Quasi-parabolic One-Resonant Biholomorphisms

In this paper we study the dynamics of germs of quasi-parabolic one-resonant biholomorphisms of ? ⁿ⁺¹ fixing the origin, namely, those germs whose differential at the origin has one eigenvalue 1 and the others having a one-dimensional family of resonant relations. We define some invariants and give conditions which ensure the existence of attracting domains for such maps.     

A universal complex structure on complete metrizable topological spaces

Let X be a topological space whose topology may be defined by a complete metric d. Taking all such metrics d we define a universal complex structure on X. For this complex structure the sheaf of germs of holomorphic functions on X coincides with the sheaf of germs of continuous functions on X, and hence the theories of topological and holomorphic vector bundles on X are the same.     

Simple tangential family germs and perestroikas of their envelopes

A tangential family is a 1-parameter system of regular curves emanating tangentially from another regular curve. We classify simple tangential family germs up to A-equivalence. We describe perestroikas of envelopes of simple tangential family germs of small codimension under small deformations of the germ among tangential families.     

C-GV triviality of function germs and Newton polyhedra

We provide estimates on the degree of C l GV determinacy ( G is one of Mather’s groups R or K ) of function germs which are defined on analytic variety V and satisfies a non-degeneracy condition with respect to some Newton polyhedron. The result gives an explicit order such that the C l geometrical structure of a function germ is preserved after higher order perturbations, which generalizes the result on C l G triviality of function germs given by M.A.S.Ruas.     

The Fatou coordinate for parabolic Dulac germs

We study the class of parabolic Dulac germs of hyperbolic polycycles. For such germs we give a constructive proof of the existence of a unique Fatou coordinate, admitting an asymptotic expansion in the power-iterated logarithm monomials.     

Holomorphic germs on certain locally convex spaces

Summary We study the bounded sets in the space of holomorphic germs defined on compact subsets of non-metrizable locally convex spaces. We relate this problem to the problem of existence of uniform Cauchy estimates for the bounded subsets. We show that the space of holomorphic germs defined on a compact subset of a reflexive dual Fréchet space is regular if the bounded subsets of the space of holomorphic germs defined at the origin have uniform Cauchy estimates.     

A family of $$ 2^{\aleph _1 } $$ logarithmic functions of distinct growth rates

We construct a totally ordered set Γ of positive infinite germs (i.e. germs of positive real-valued functions that tend to +∞), with order type being the lexicographic product ℵ₁ × ℤ². We show that Γ admits $ 2^{\aleph _1 } $ 2^{\aleph _1 } order preserving automorphisms of pairwise distinct growth rates.     

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 ³ 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 ³. Finally, we find in higher embedding dimension three different families with these same properties. Partially supported by DGICYT, BFM2002-04797 and HPRN-CT-2001-00271 Mathematical Subject Classification (2000): 11E25, 14P15.An erratum to this article can be found at     

The Oscillation of Elliptic Systems

Foliated differential forms were introduced in [7], [9], to study the cohomology on a RIEMANNian foliated manifold with coefficients in the sheaf of germs of foliated differential forms. In this paper the notion of DE RHAM like current of the type (p, q) is defined for a RIEMANNian foliated manifold and some properties of various differential operators acting on the spaces of currents are given. In particular, special DE RHAM like currents are considered namely the foliated ones. It turns out that the space of foliated p-forms is dense in the space of foliated p-currents with the usual topology. We get certain results concerning the cohomology on a RIEMANNian foliated manifold with coefficients in the sheaf of germs of foliated currents.     

Equivariant Poincaré series of filtrations and topology

Earlier, for an action of a finite group G on a germ of an analytic variety, an equivariant G-Poincaré series of a multi-index filtration in the ring of germs of functions on the variety was defined as an element of the Grothendieck ring of G-sets with an additional structure. We discuss to which extent the G-Poincaré series of a filtration defined by a set of curve or divisorial valuations on the ring of germs of analytic functions in two variables determines the (equivariant) topology of the curve or of the set of divisors.     

Formal and finite order equivalences

We show that two families of germs of real-analytic subsets in ${{\mathbb C}^{n}}$ are formally equivalent if and only if they are equivalent of any finite order. We further apply the same technique to obtain analogous statements for equivalences of real-analytic self-maps and vector fields under conjugations. On the other hand, we provide an example of two sets of germs of smooth curves that are equivalent of any finite order but not formally equivalent.     

The link of the germ of a semi-algebraic metric space

In this paper we investigate the metric properties of semi-algebraic germs. More precisely we introduce a counterpart to the notion of link for semi-algebraic metric spaces, which is often used to study the topology. We prove that it totally determines the metric type of the germ. We give a nice consequence for semi-algebraically bi-Lipschitz homeomorphic semi-algebraic germs.