Equisingular Approximation of Analytic Germs

Aftab Yusuf Patel, The University of Western Ontario

Abstract

This thesis deals with the problem of approximating germs of real or complex analytic spaces by Nash or algebraic germs. In particular, we investigate the problem of approximating analytic germs in various ways while preserving the Hilbert-Samuel function, which is of importance in the resolution of singularities. We first show that analytic germs that are complete intersections can be arbitrarily closely approximated by algebraic germs which are complete intersections with the same Hilbert-Samuel function. We then show that analytic germs whose local rings are Cohen-Macaulay can be arbitrarily closely approximated by Nash germs whose local rings are Cohen- Macaulay and have the same Hilbert-Samuel function. Finally we prove that we may approximate arbitrary analytic germs by topologically equisingular Nash germs which have the same Hilbert-Samuel function.