Maximal and Typical Topology of Real Polynomial Singularities

Given a polynomial map $\psi:S^m\to \mathbb{R}^k$ with components of degree $d$, we investigate the structure of the semialgebraic set $Z\subseteq S^m$ consisting of those points where $\psi$ and its derivatives satisfy a given list of polynomial equalities and inequalities (we call such a set a "singularity"). Concerning the upper estimate on the topological complexity of a polynomial singularity, we sharpen the classical bound $b(Z)\leq O(d^{m+1})$, proved by Milnor, with \begin{equation}\label{eq:abstract} b(Z)\leq O(d^{m}),\end{equation} which holds for the generic polynomial map. For what concerns the "lower bound" on the topology of $Z$, we prove a general semicontinuity result for the Betti numbers of the zero set of $\mathcal{C}^0$ perturbations of smooth maps -- the case of $\mathcal{C}^1$ perturbations is the content of Thom's Isotopy Lemma (essentially the Implicit Function Theorem). This result is of independent interest and it is stated for general maps (not just polynomial); this result implies that small continuous perturbations of $\mathcal{C}^1$ manifolds have a richer topology than the one of the original manifold. We then compare the extremal case with a random one and prove that on average the topology of $Z$ behaves as the "square root" of its upper bound: for a random Kostlan map $\psi:S^m\to \mathbb{R}^k$ with components of degree $d$, we have: \begin{equation} \mathbb{E}b(Z)=\Theta(d^{\frac{m}{2}}).\end{equation} This generalizes classical results of Edelman-Kostlan-Shub-Smale from the zero set of a random map, to the structure of its singularities.