{"paper":{"title":"Low degree approximation of random polynomials","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.PR"],"primary_cat":"math.AG","authors_text":"Antonio Lerario, Daouda Niang Diatta","submitted_at":"2018-12-25T17:05:30Z","abstract_excerpt":"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 $S^n$. 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 p"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1812.10137","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}