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JOURNALS // Moscow Mathematical Journal // Archive

Mosc. Math. J., 2003 Volume 3, Number 3, Pages 1097–1112 (Mi mmj123)

This article is cited in 3 papers

On the topology of singularities of Maxwell sets

V. D. Sedykh

Gubkin Russian State University of Oil and Gas

Abstract: We determine new conditions for the coexistence of corank-one singularities of the Maxwell set of a generic family of smooth functions with respect to taking global minima (or maxima) in cases when this set does not have more complicated singularities. In particular, the Euler number of every odd-dimensional manifold of singularities of a given type is a linear combination of the Euler numbers of even-dimensional manifolds of singularities of higher codimensions. The coefficients of this combination are universal numbers (that is, they do not depend on the family and depend only on the classes of singularities).
We obtain these conditions as a corollary to the general coexistence conditions for corank 1 singularities of generic wave fronts which were found recently by the author. As an application, we obtain many-dimensional generalizations of the classical Bose formula relating the number of supporting curvature circles for a smooth closed convex generic plane curve to the number of supporting circles which are tangent to this curve at three points.

Key words and phrases: Families of smooth functions, global minima and maxima, Maxwell sets, corank-one singularities of smooth functions, Euler number, convex curves, supporting hyperspheres.

MSC: Primary 58C05, 58K30; Secondary 53A04

Received: June 26, 2002

Language: English

DOI: 10.17323/1609-4514-2003-3-3-1097-1112



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