2020/04/28 by Thaís M. Dalbelo, Dalbelo, Thaís M., Hellen Santana +1
Computer Science · Mathematics · #Algebraic Geometry and Number Theory #Commutative Algebra and Its Applications #FOS: Mathematics #Geometric Topology (math.GT) #Polynomial and algebraic computation
paper · pdf · doi:10.48550/arxiv.2004.13793
openalex publication_date 2020/04/28 · openalex created_date 2020/05/13 · openalex updated_date 2026/07/28
The generalization of the Morse theory presented by Goresky and MacPherson is a landmark that divided completely the topological and geo\-me\-tri\-cal study of singular spaces. Let \Xt\t be a suitable family of germs at 0 of complete intersection varieties in ℂn and \ft\t, \gt\t families of non-constant polynomial functions on Xt. If the germs Xt, Xt ∩ ft-1(0) and Xt∩ ft-1(0) ∩ gt-1(0) are non-degenerate, locally tame, complete intersection varieties, for each t, we prove that the difference of the Brasselet numbers, \rm Bft,Xt(0) and \rm Bft,Xt∩ gt-1(0)(0), is related with the number of Morse critical points on the regular part of the Milnor fiber of ft appearing in a morsefication of gt, even in the case where gt has a critical locus with arbitrary dimension. This result connects topological and geometric properties and allows us to determine some interesting formulae, mainly in terms of the combinatorial information from Newton polyhedra.