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BILIPSCHITZ INVARIANCE OF THE MULTIPLICITY

Published online by Cambridge University Press:  01 March 1997

JEAN-JACQUES RISLER
Affiliation:
École Normale Supérieure, 45 Rue d'Ulm, 75005 Paris, France
DAVID TROTMAN
Affiliation:
Unité de Recherche Associée au CNRS 225, Université de Provence, 39 Rue Joliot-Curie, 13453 Marseille, France
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Abstract

A famous problem of Zariski, stated in 1971 [21], is to decide whether the classical algebro-geometric multiplicity mo(f) of a complex hypersurface f−1(0) at an isolated singularity O is an invariant of the topological type of the embedded germ. That is, if f[ratio ]Cn, OC, 0 and g[ratio ]Cn, OC, 0 are two germs of polynomial functions, and there is a homeomorphism ϕ defined between two open neighbourhoods U and V of O in Cn such that ϕ(U) = V, ϕ(Uf−1(0)) = Vg−1(0) and ϕ(O) = O, then is it always the case that mo(f) = mo(g)? Here the multiplicity mo(f) is defined to be the intersection number of f−1(0) with a generic complex line L passing through O in Cn, that is, the number of points of intersection of f−1(0) with a line which is an arbitrarily small perturbation of L. If f is reduced, then mo(f) is equal to the order of the polynomial f(z) at O.

Type
Research Article
Copyright
© The London Mathematical Society 1997

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