Get 20M+ Full-Text Papers For Less Than $1.50/day. Start a 14-Day Trial for You or Your Team.

Learn More →

On the Canonical Stratification of Complex Analytic Functions

On the Canonical Stratification of Complex Analytic Functions ON THE CANONICAL STRATIFICATION OF COMPLEX ANALYTIC FUNCTIONS J. W. BRUCE In the Thorn-Mather proof of the density of topologically stable mappings Whitney stratifications of large parts of the jet spaces J\n, p) are constructed (see [4]). These stratifications yield a topological classification of germs of finite singularity type which satisfy a natural multitransversality condition with respect to the given stratification, for two such germs whose jets are in the same connected stratum are topologically equivalent. The construction of the stratifications mentioned above essentially reduces to the problem of finding a minimal Whitney stratification of a versal unfolding of each jet of finite singularity type. However apart from the rather easy case of simple singularities [2] the problem of determining this minimal stratification even for complex analytic functions with isolated singularity is very difficult. Indeed the answer is not known even for the simple elliptic families E , k = 6, 7, 8. Using results of Hironaka [6] we give a necessary (numerical) condition for a given family of complex analytic functions with isolated singularities to constitute a canonical stratum; namely the number of cusps (see [8]) must be constant along the family. Examples of Greuel [5] then furnish http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Bulletin of the London Mathematical Society Wiley

On the Canonical Stratification of Complex Analytic Functions

Download PDF
4 pages

Loading next page...
 
/lp/wiley/on-the-canonical-stratification-of-complex-analytic-functions-Ji84eYCw8R

References (0)

References for this paper are not available at this time. We will be adding them shortly, thank you for your patience.

Publisher
Wiley
Copyright
© London Mathematical Society
ISSN
0024-6093
eISSN
1469-2120
DOI
10.1112/blms/12.2.111
Publisher site
See Article on Publisher Site

Abstract

ON THE CANONICAL STRATIFICATION OF COMPLEX ANALYTIC FUNCTIONS J. W. BRUCE In the Thorn-Mather proof of the density of topologically stable mappings Whitney stratifications of large parts of the jet spaces J\n, p) are constructed (see [4]). These stratifications yield a topological classification of germs of finite singularity type which satisfy a natural multitransversality condition with respect to the given stratification, for two such germs whose jets are in the same connected stratum are topologically equivalent. The construction of the stratifications mentioned above essentially reduces to the problem of finding a minimal Whitney stratification of a versal unfolding of each jet of finite singularity type. However apart from the rather easy case of simple singularities [2] the problem of determining this minimal stratification even for complex analytic functions with isolated singularity is very difficult. Indeed the answer is not known even for the simple elliptic families E , k = 6, 7, 8. Using results of Hironaka [6] we give a necessary (numerical) condition for a given family of complex analytic functions with isolated singularities to constitute a canonical stratum; namely the number of cusps (see [8]) must be constant along the family. Examples of Greuel [5] then furnish

Journal

Bulletin of the London Mathematical Society – Wiley

Published: Mar 1, 1980

There are no references for this article.