Distance-Genericity for Real Algebraic Hypersurfaces

• Published in: Canadian journal of mathematics 1984-04, Vol.36 (2), p.374-384
• Main Authors: Bruce, J. W., Gibson, C. G.
• Format: Article
• Language: English
• ISSN: 0008-414X; 1496-4279
• DOI: 10.4153/CJM-1984-023-0

One of the original applications of catastrophe theory envisaged by Thom was that of discussing the local structure of the focal set for a (generic) smooth submanifold M ⊆ R n + 1. Thom conjectured that for a generic M there would be only finitely many local topological models, a result proved by Looijenga in [4]. The objective of this paper is to extend Looijenga's result from the smooth category to the algebraic category (in a sense explained below), at least in the case when M has codimension 1. Looijenga worked with the compactified family of distance-squared functions on M (defined below), thus including the family of height functions on M whose corresponding catastrophe theory yields the local structure of the focal set at infinity. For the family of height functions the appropriate genericity theorem in the smooth category was extended to the algebraic case in [1], so that the present paper can be viewed as a natural continuation of the first author's work in this direction.