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Modified Nash triviality of a family of zero-sets of real polynomial mappings

Toshizumi Fukui, Satoshi Koike, Masahiro Shiota (1998)

Annales de l'institut Fourier

In this paper we introduce the notion of modified Nash triviality for a family of zero sets of real polynomial map-germs as a desirable one. We first give a Nash isotopy lemma which is a useful tool to show triviality.Then, using it, we prove two types of modified Nash triviality theorem and a finite classification theorem for this triviality. These theorems strengthen similar topological results.

Modules pour les familles de courbes planes

Jean-Paul Dufour (1989)

Annales de l'institut Fourier

L’étude des familles de courbes plane différentiables se ramène a celle des diagrammes ℝ ← f S → σ ℝ 2 où S est une surface, f et σ étant différentiables. Dans la classification de ces diagrammes à équivalence près il apparaît trois types de modules: des modules locaux attachés à chaque fronce de σ , des modules semi-locaux attachés à la superposition en un même point de plusieurs situations locales, des modules globaux attachés aux “courbes de contact” le long desquelles certaines courbes sont tangentes. Nous explicitons...

Monge-Ampère equations and surfaces with negative Gaussian curvature

Mikio Tsuji (1997)

Banach Center Publications

In [24], we studied the singularities of solutions of Monge-Ampère equations of hyperbolic type. Then we saw that the singularities of solutions do not coincide with the singularities of solution surfaces. In this note we first study the singularities of solution surfaces. Next, as the applications, we consider the singularities of surfaces with negative Gaussian curvature. Our problems are as follows: 1) What kinds of singularities may appear?, and 2) How can we extend the surfaces beyond the singularities?...

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