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.

我们证明了“高概率”随机Kostlan多项式\（N + 1 \）和学位的许多变量d可以通过多项式“低度”的逼近，而不改变其零集的拓扑结构上球体\（ \ mathbb {S} ^ n \）。逼近度的“低级”与“高概率”之间的相关性是定量的：例如，以压倒性的概率，度数d的科斯特兰多项式的零集与度数\（（ O（\ sqrt {d \ log d}）\）。证明基于\（C ^ 1 \）大小的概率研究稳定的Kostlan多项式的邻域。作为推论，我们证明某些拓扑类型（例如，具有深嵌套的椭圆形曲线或具有丰富拓扑的超曲面）以零集形式出现为随机的Kostlan多项式的零集合。