C1 preservation of multiplicity

Ephraim, Robert

doi:10.1215/s0012-7094-76-04361-1

Cited by 17 publications

(10 citation statements)

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“…Y.-N. Gau and J. Lipman [2] have proved the Zariski conjecture on hypersuface multiplicity even in the non-hypersurface case under the assumption that the homeomorphism is a weak diffeomorphism. The hypersurface case was implicitly shown in [1].…”

Section: Resultsmentioning

confidence: 99%

(SSP) geometry with directional homeomorphisms

Koike¹,

Păunescu²

2015

jsing

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In a previous paper [6] we discussed several directional properties of sets satisfying the sequence selection property, denoted by (SSP) for short, and developed the (SSP) geometry via bi-Lipschitz transformations. In this paper we introduce the notion of directional homeomorphism and show that we can develop also the (SSP) geometry with directional transformations. For many important results proved in [6] for bi-Lipschitz homeomorphisms we describe the analogues for directional homeomorphisms as well.

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Section: Resultsmentioning

confidence: 99%

(SSP) geometry with directional homeomorphisms

Koike¹,

Păunescu²

2015

jsing

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“…The answer is, nevertheless, known to be yes in several special cases the list of which can be found in the recent first author's survey article [3]. In particular, Ephraim [2] proved that multiplicity is preserved by ambient C 1 -diffeomorphisms; his paper inspired some of our proofs. In this section, we give a partial positive answer to Zariski's question in the special case of 'small' homeomorphisms for Newton nondegenerate isolated singularities and one-parameter families of isolated singularities.…”

Section: Applications To Zariski's Multiplicity Questionmentioning

confidence: 97%

“…[2]). In particular, there exists a closed disc D ⊆ L around 0 such that, for any closed disc D ⊆ D around 0,…”

Section: Multiplicity and Rouché Satellitesmentioning

confidence: 99%

Multiplicity of complex hypersurface singularities, Rouché satellites and Zariski's problem

Eyral

Gasparim

2007

Comptes Rendus. Mathématique

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Let f, g : (C n , 0) → (C, 0) be reduced germs of holomorphic functions. We show that f and g have the same multiplicity at 0, if and only if, there exist reduced germs f and g analytically equivalent to f and g, respectively, such that f and g satisfy a Rouché type inequality with respect to a generic 'small' circle around 0. As an application, we give a reformulation of Zariski's multiplicity question and a partial positive answer to it. To cite this article: C. Eyral, E. Gasparim, C. R. Acad. Sci. Paris, Ser. I 344 (2007).

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“…answer when n = 2. For any n, Ephraim in [5] and independently Trotman in [14] showed that the Zariski's problem has a positive answer if the homeomorphism ϕ and its inverse are C 1 . Since the notion of multiplicity is defined for any complex analytic set with pure dimension (see, for example, [3] for a definition of multiplicity in higher codimension), we can get the same Zariski's problem in any codimension.…”

Section: Introductionmentioning

confidence: 99%