Doctor of Philosophy
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.
Summary for Lay Audience
Replacing geometric objects with simpler ones that share some properties with the original is an important operation in many branches of geometry. When the simpler replacement can be derived from the original by stopping a limiting process, it is called an approximation. In this thesis we deal with the local approximations of real or complex analytic spaces which preserve various properties. “Local” here indicates that we are interested in approximations near a point as opposed to global approximations that are approximations over some finite region of space. Analytic spaces near a point are defined by finite sets of power series. We look for approximations that preserve (i) the algebro-geometric class of the original, i.e., Complete Intersection or Cohen-Macaulay or (ii) the topological type of the original. In both cases we also impose the requirement that the approximants have the same Hilbert-Samuel function as the original. This additional constraint is motivated by the fact that the Hilbert-Samuel function is thought of as a measure of how singular an analytic space is near a point and plays an important role in Hironaka’s seminal work on desingularization of analytic spaces. We approximate the original analytic space by approximating the power series that define it near a point. We show that it is possible to find approximations of the form (i) and (ii) whose defining power series belong to an algebraically simpler class than those of the original. Specifically, in the case of Complete Intersections we show that we can approximate the original set of defining power series by polynomials. In the other cases we show that the approximating power series can be chosen to be Nash, i.e., power series satisfying a polynomial equation.
Patel, Aftab Yusuf, "Equisingular Approximation of Analytic Germs" (2021). Electronic Thesis and Dissertation Repository. 8261.
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