Iterations and logarithms of formal automorphisms. (Short Communication).
K-subanalytic rectilinearization and uniformization
We prove rectilinearization and uniformization theorems for K-subanalytic (∝anK-definable) sets and functions using the Lion-Rolin formula. Parallel reasoning gives standard results for the subanalytic case.
Local analytic invariants
Local analytic rings
Lokale Algebra und lokale Geometrie.
Nicht-ausgeartete reguläre Überlagerungen.
Note on convergence and orders of vanishing along curves.
On analytic models of synthetic differential geometry
On gradients of functions definable in o-minimal structures
We prove the o-minimal generalization of the Łojasiewicz inequality , with , in a neighborhood of , where is real analytic at and . We deduce, as in the analytic case, that trajectories of the gradient of a function definable in an o-minimal structure are of uniformly bounded length. We obtain also that the gradient flow gives a retraction onto levels of such functions.
On microlocal -function
On some noetherian rings of germs on a real closed field
Let R be a real closed field, and denote by the ring of germs, at the origin of Rⁿ, of functions in a neighborhood of 0 ∈ Rⁿ. For each n ∈ ℕ, we construct a quasianalytic subring with some natural properties. We prove that, for each n ∈ ℕ, is a noetherian ring and if R = ℝ (the field of real numbers), then , where ₙ is the ring of germs, at the origin of ℝⁿ, of real analytic functions. Finally, we prove the Real Nullstellensatz and solve Hilbert’s 17th Problem for the ring .
On the Induced h-Structure on an Open Subset of the Rigid Analytic Space IPl (k) .
On the rings of formal solutions of polynomial differential equations
The paper establishes the basic algebraic theory for the Gevrey rings. We prove the Hensel lemma, the Artin approximation theorem and the Weierstrass-Hironaka division theorem for them. We introduce a family of norms and we look at them as a family of analytic functions defined on some semialgebraic sets. This allows us to study the analytic and algebraic properties of this rings.
Opérateurs différentiels sur des cônes normaux de dimension 2
Opérateurs différentiels sur les espaces analytiques.
Préparation des fonctions sous-analytiques globales et lieu d'analyticité
We give a new proof of Kurdyka-Tamm's theorem on the analytic locus of a subanalytic function.
Preparation theorems for matrix valued functions
We generalize the Malgrange preparation theorem to matrix valued functions satisfying the condition that vanishes to finite order at . Then we can factor near (0,0), where is inversible and is polynomial function of depending on . The preparation is (essentially) unique, up to functions vanishing to infinite order at , if we impose some additional conditions on . We also have a generalization of the division theorem, and analytic versions generalizing the Weierstrass preparation...
Quotienten differenzierbarer und komplexer Räume nach eigentlich-diskontinuierlichen Gruppen.
Radially symmetric plurisubharmonic functions
In this note we consider radially symmetric plurisubharmonic functions and the complex Monge-Ampère operator. We prove among other things a complete characterization of unitary invariant measures for which there exists a solution of the complex Monge-Ampère equation in the set of radially symmetric plurisubharmonic functions. Furthermore, we prove in contrast to the general case that the complex Monge-Ampère operator is continuous on the set of radially symmetric plurisubharmonic functions. Finally...
Reduktive Automorphismengruppen analytischer C-Algebren.