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The dependence between the \u201clow degree\u201d of the approximation and the \u201chigh probability\u201d is quantitative: for example, with overwhelming probability, the zero set of a Kostlan polynomial of degree <jats:italic>d<\/jats:italic> is isotopic to the zero set of a polynomial of degree <jats:inline-formula><jats:alternatives><jats:tex-math>$$O(\\sqrt{d \\log d})$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mi>O<\/mml:mi>\n                    <mml:mo>(<\/mml:mo>\n                    <mml:msqrt>\n                      <mml:mrow>\n                        <mml:mi>d<\/mml:mi>\n                        <mml:mo>log<\/mml:mo>\n                        <mml:mi>d<\/mml:mi>\n                      <\/mml:mrow>\n                    <\/mml:msqrt>\n                    <mml:mo>)<\/mml:mo>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula>. The proof is based on a probabilistic study of the size of <jats:inline-formula><jats:alternatives><jats:tex-math>$$C^1$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:msup>\n                    <mml:mi>C<\/mml:mi>\n                    <mml:mn>1<\/mml:mn>\n                  <\/mml:msup>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula>-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.\n<\/jats:p>","DOI":"10.1007\/s10208-021-09506-y","type":"journal-article","created":{"date-parts":[[2021,3,25]],"date-time":"2021-03-25T22:43:10Z","timestamp":1616712190000},"page":"77-97","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":11,"title":["Low-Degree Approximation of Random Polynomials"],"prefix":"10.1007","volume":"22","author":[{"given":"Daouda Niang","family":"Diatta","sequence":"first","affiliation":[]},{"given":"Antonio","family":"Lerario","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2021,3,25]]},"reference":[{"key":"9506_CR1","unstructured":"Michele Ancona. 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