2016/07/08 by Alberti, Lionel
#32B10 #32C05 #32S30 #49Q15 #58K60 #58K65 #Algebraic Geometry (math.AG) #FOS: Mathematics #Metric Geometry (math.MG)
paper · doi:10.48550/arxiv.1607.02544
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).