Statistics of topological structures of smooth functions, of polynomials, of trigonometric polynomials, of affine Coxeter groups and the Sixteenth Problem of Hilbert. (Lecture 1) Series of Lectures in "Experimental Discoveries of Mathematical Facts"
Prof. V.I. Arnold
(Steklov Mathematical Institute, Moscow, Russia)
One associates to a smooth function a graph, defined as the topological space whose points are the connected components of the level hypersurfaces of the function. The talk describes the structures of these graphs for the Morse functions with a fixed number of critical points and for the polynomials and trigonometric polynomials of given degree or a given spectrum (provided, for instance, by the Newton polyhedron of that trigonometric polynomial for an affine system of roots).
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