Topological finite-determinacy of functions with non-isolated singularities

Details Topological finite-determinacy of functions with non-isolated singularities Topological finite-determinacy of functions with non-isolated singularities

2002-10-17

arxiv.org/abs/math/0210266v2

Mathematics

Algebraic Geometry

26 pages; Main Theorem improved

Scientific paper

We introduce the concept of topological finite-determinacy for germs of analytic functions within a fixed ideal $I$, which provides a notion of topological finite-determinacy of functions with non-isolated singularities. We prove the following statement which generalizes classical results of Thom and Varchenko: let $A$ be the complement in the ideal $I$ of the space of germs whose topological type remains unchanged under a deformation within the ideal that only modifies sufficiently large order terms of the Taylor expansion; then $A$ has infinite codimension in $I$ in a suitable sense. We also prove the existence of generic topological types of families of germs of $I$ parametrized by an irreducible analytic set.

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