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 $$\mathbb {S}^n$$ . 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.