Polynomial Bound on the Local Betti Numbers of a Real Analytic Germ

Abstract

This article proves the existence of a bound on the sum of local Betti numbers of a real analytic germ by a polynomial function of its multiplicity. This result can be interpreted as a localization of the classical Oleinik-Petrovsky bound (also known as Thom-Milnor bound) on the sum of Betti numbers of a semi-algebraic set. The proof relies on an interplay between geometric and algebraic arguments whose key elements are the tangent cone of the germ, the Thom-Mather topological trivialization theorem, the Oleinik-Petrovsky bound, and a result by D. Mumford and J. Heintz bounding the degrees of the generators of an ideal by a polynomial function of the geometric degree of its associated variety. Our result is then applied to yield bounds on invariants from singularity theory, such as the Lipschitz-Killing curvature invariants and the Vitushkin variations (which include the local density of a germ).