The probabilistic method in real singularity theory

TL;DR

This paper demonstrates how probabilistic techniques can establish the existence of real polynomial singularities and hypersurfaces with complex topological and geometric features, advancing understanding in real algebraic geometry.

Contribution

It introduces probabilistic methods to prove the existence of real algebraic varieties with rich topology and geometric structures, which was previously challenging.

Findings

Existence of real polynomial singularities with maximal Betti numbers

Construction of real hypersurfaces with numerous umbilical points

Application of probabilistic methods to real algebraic geometry

Abstract

We explain how to use the probabilistic method to prove the existence of real polynomial singularities with rich topology, i.e. with total Betti number of the maximal possible order. We show how similar ideas can be used to produce real algebraic projective hypersurfaces with a rich structure of umbilical points.