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.
Cite this article
Javier Fernández de Bobadilla, Topological finite-determinacy of functions with non-isolated singularities. Comment. Math. Helv. 79 (2004), no. 4, pp. 659–688DOI 10.1007/S00014-004-0813-1