{"paper":{"title":"Point-curve incidences in the complex plane","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Adam Sheffer, Endre Szab\\'o, Joshua Zahl","submitted_at":"2015-02-24T23:18:50Z","abstract_excerpt":"We prove an incidence theorem for points and curves in the complex plane. Given a set of $m$ points in ${\\mathbb R}^2$ and a set of $n$ curves with $k$ degrees of freedom, Pach and Sharir proved that the number of point-curve incidences is $O\\big(m^{\\frac{k}{2k-1}}n^{\\frac{2k-2}{2k-1}}+m+n\\big)$. We establish the slightly weaker bound $O_\\varepsilon\\big(m^{\\frac{k}{2k-1}+\\varepsilon}n^{\\frac{2k-2}{2k-1}}+m+n\\big)$ on the number of incidences between $m$ points and $n$ (complex) algebraic curves in ${\\mathbb C}^2$ with $k$ degrees of freedom. We combine tools from algebraic geometry and differe"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1502.07003","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"}