Low degree approximation of random polynomials

We prove that with "high probability" a random Kostlan polynomial in $n+1$ many variables and of degree $d$ can be approximated by a polynomial of "low degree" without changing the topology of its zero set on the sphere $S^n$. The dependence between the "low degree" of the approximation and the "high probability" is quantitative: for example, with overwhelming probability the zero set of a Kostlan polynomial of degree $d$ is isotopic to the zero set of a polynomial of degree $O(\sqrt{d \log d})$. The proof is based on a probabilistic study of the size of $C^1$-stable neighborhoods of Kostlan polynomials. As a corollary we prove that certain topological types (e.g. curves with deep nests of ovals or hypersurfaces with rich topology) have exponentially small probability of appearing as zero sets of random Kostlan polynomials.