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Foundations of Computational Mathematics
Published in: Foundations of Computational Mathematics 3/2013

01-06-2013

Isosingular Sets and Deflation

Authors: Jonathan D. Hauenstein, Charles W. Wampler

Published in: Foundations of Computational Mathematics | Issue 3/2013

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Abstract

This article introduces the concept of isosingular sets, which are irreducible algebraic subsets of the set of solutions to a system of polynomial equations constructed by taking the closure of points with a common singularity structure. The definition of these sets depends on deflation, a procedure that uses differentiation to regularize solutions. A weak form of deflation has proven useful in regularizing algebraic sets, making them amenable to treatment by the algorithms of numerical algebraic geometry. We introduce a strong form of deflation and define deflation sequences, which, in a different context, are the sequences arising in Thom–Boardman singularity theory. We then define isosingular sets in terms of deflation sequences. We also define the isosingular local dimension and examine the properties of isosingular sets. While isosingular sets are of theoretical interest as constructs for describing singularity structures of algebraic sets, they also expand the kinds of algebraic set that can be investigated with methods from numerical algebraic geometry.
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Appendix
Available only for authorised users
Footnotes
1
Constructible algebraic sets are sets formed from algebraic sets with a finite number of Boolean operations (union, intersection, and complementation). The closure of such sets are the same in both the Zariski topology and the usual complex topology, so we draw no distinction in this article.
 
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Metadata
Title
Isosingular Sets and Deflation
Authors
Jonathan D. Hauenstein
Charles W. Wampler
Publication date
01-06-2013
Publisher
Springer-Verlag
Published in
Foundations of Computational Mathematics / Issue 3/2013
Print ISSN: 1615-3375
Electronic ISSN: 1615-3383
DOI
https://doi.org/10.1007/s10208-013-9147-y

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