Maximal and Typical topology of Real Polynomial Singularities

November 14, 2019
Versione stampabile

Luogo: Povo Zero, via Sommarive 14 (Povo) - Seminar Room "-1"
Ore: 3.00 PM

"Doc in Progress" is pleased to introduce you to

  • Michele Stecconi - PhD in Mathematics, Scuola Superiore di Studi Avanzati - SISSA

Abstract:
In this talk I will consider maps from the m-sphere to R^k, defined by k homogenous polynomials of degree d in m+1 variables. I will investigate the topology of set Z of points where those maps and their derivatives (up to order r) satisfy a given list of polynomial equalities and inequalities. I call such set of conditions a "singularity class" and Z a ''singularity". Examples are:  the zero set (here no derivative is involved, i.e. r=0); the set of critical points (r=1); the set of critical points with a prescribed signature of the hessian (k=1,r=2).I will present the content of a homonymous paper (a joint work with with Antonio Lerario, on arxiv) concerning the asymptotic of the Betti numbers of Z as d grows, within a fixed "singularity class". First, for what regards the maximal topology, I will try to convince you that it is possible to find maps for which Z has arbitrarily large Betti numbers (this is subtle when derivatives are involved in the singularity class), however there is a constraint on the degree: b(Z)=O(d^m).  On the other hand, the typical behaviour of the Betti numbers is ~d^m/2 (the square root of the maximal). The word "typical" means that I am taking the expected value with respect to a certain natural probability on the space of polynomials, called Kostlan measure.