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• W2906334755 abstract "We prove that with high a random Kostlan polynomial in $n+1$ many variables and of $d$ 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 is quantitative: for example, with overwhelming probability the zero set of a Kostlan polynomial of $d$ is isotopic to the zero set of a polynomial of $O(sqrt{d log d})$. 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." @default.
• W2906334755 created "2019-01-01" @default.
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• W2906334755 date "2018-12-25" @default.
• W2906334755 modified "2023-10-01" @default.
• W2906334755 title "Low degree approximation of random polynomials" @default.
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