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 Sn. 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(dlogd−−−−−√). The proof is based on a probabilistic study of the size of C1-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.

Low-Degree Approximation of Random Polynomials / Diatta, Daouda Niang; Lerario, Antonio. - In: FOUNDATIONS OF COMPUTATIONAL MATHEMATICS. - ISSN 1615-3375. - 22:1(2022), pp. 77-97. [10.1007/s10208-021-09506-y]

Low-Degree Approximation of Random Polynomials

Diatta, Daouda Niang;Lerario, Antonio
2022

Abstract

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 Sn. 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(dlogd−−−−−√). The proof is based on a probabilistic study of the size of C1-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.
22
1
77
97
https://doi.org/10.1007/s10208-021-09506-y
https://arxiv.org/abs/1812.10137
Diatta, Daouda Niang; Lerario, Antonio
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Utilizza questo identificativo per citare o creare un link a questo documento: http://hdl.handle.net/20.500.11767/126823
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