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On condition ( a f ) of a stratified mapping

Satoshi Koike — 1983

Annales de l'institut Fourier

For a stratified mapping f , we consider the condition ( a f ) concerning the kernel of the differential of f . We show that the condition ( a f ) is equivalent to the condition ( a f S ) which has a more obvious geometric content.

The directional dimension of subanalytic sets is invariant under bi-Lipschitz homeomorphisms

Satoshi KoikeLaurentiu Paunescu — 2009

Annales de l’institut Fourier

Let A n be a set-germ at 0 n such that 0 A ¯ . We say that r S n - 1 is a direction of A at 0 n if there is a sequence of points { x i } A { 0 } tending to 0 n such that x i x i r as i . Let D ( A ) denote the set of all directions of A at 0 n . Let A , B n be subanalytic set-germs at 0 n such that 0 A ¯ B ¯ . We study the problem of whether the dimension of the common direction set, dim ( D ( A ) D ( B ) ) is preserved by bi-Lipschitz homeomorphisms. We show that although it is not true in general, it is preserved if the images of A and B are also subanalytic. In particular...

Motivic-type invariants of blow-analytic equivalence

Satoshi KoikeAdam Parusiński — 2003

Annales de l'Institut Fourier

To a given analytic function germ f : ( d , 0 ) ( , 0 ) , we associate zeta functions Z f , + , Z f , - [ [ T ] ] , defined analogously to the motivic zeta functions of Denef and Loeser. We show that our zeta functions are rational and that they are invariants of the blow-analytic equivalence in the sense of Kuo. Then we use them together with the Fukui invariant to classify the blow-analytic equivalence classes of Brieskorn polynomials of two variables. Except special series of singularities our method classifies as well the blow-analytic...

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

Toshizumi FukuiSatoshi KoikeMasahiro 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.

Directional properties of sets definable in o-minimal structures

Satoshi KoikeTa Lê LoiLaurentiu PaunescuMasahiro Shiota — 2013

Annales de l’institut Fourier

In a previous paper by Koike and Paunescu, it was introduced the notion of direction set for a subset of a Euclidean space, and it was shown that the dimension of the common direction set of two subanalytic subsets, called , is preserved by a bi-Lipschitz homeomorphism, provided that their images are also subanalytic. In this paper we give a generalisation of the above result to sets definable in an o-minimal structure on an arbitrary real closed field. More precisely, we first prove our main theorem...

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