{"paper":{"title":"Primary singularities of vector fields on surfaces","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DS","authors_text":"Francisco-Javier Turiel, Morris W. Hirsch","submitted_at":"2018-07-12T10:50:47Z","abstract_excerpt":"Unless another thing is stated one works in the $C^\\infty$ category and manifolds have empty boundary. Let $X$ and $Y$ be vector fields on a manifold $M$. We say that $Y$ tracks $X$ if $[Y,X]=fX$ for some continuous function $f\\colon M\\rightarrow\\mathbb R$. A subset $K$ of the zero set ${\\mathsf Z}(X)$ is an essential block for $X$ if it is non-empty, compact, open in ${\\mathsf Z}(X)$ and its Poincar\\'e-Hopf index does not vanishes. One says that $X$ is non-flat at $p$ if its $\\infty$-jet at $p$ is non-trivial. A point $p$ of ${\\mathsf Z}(X)$ is called a primary singularity of $X$ if any vecto"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1807.04533","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"}