{"paper":{"title":"Persistence of Zero Sets","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.CG","math.GT"],"primary_cat":"math.AT","authors_text":"Marek Kr\\v{c}\\'al, Peter Franek","submitted_at":"2015-07-11T20:07:40Z","abstract_excerpt":"We study robust properties of zero sets of continuous maps $f:X\\to\\mathbb{R}^n$. Formally, we analyze the family $Z_r(f)=\\{g^{-1}(0):\\,\\,\\|g-f\\|<r\\}$ of all zero sets of all continuous maps $g$ closer to $f$ than $r$ in the max-norm. The fundamental geometric property of $Z_r(f)$ is that all its zero sets lie outside of $A:=\\{x:\\,|f(x)|\\ge r\\}$. We claim that once the space $A$ is fixed, $Z_r(f)$ is \\emph{fully} determined by an element of a so-called cohomotopy group which---by a recent result---is computable whenever the dimension of $X$ is at most $2n-3$. More explicitly, the element is a h"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1507.04310","kind":"arxiv","version":4},"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"}