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

TL;DR

This paper establishes a polynomial bound on the sum of local Betti numbers of a real analytic germ, linking algebraic and geometric properties to classical bounds and applying these to invariants in singularity theory.

Contribution

It introduces a polynomial bound on local Betti numbers of real analytic germs, extending classical bounds and connecting algebraic and geometric invariants.

Findings

Bound on local Betti numbers by polynomial function of multiplicity

Extension of Oleinik-Petrovsky bound to local invariants

Application to singularity invariants like Lipschitz-Killing curvature

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).