{"paper":{"title":"Semiarcs with long secants","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Bence Csajb\\'ok","submitted_at":"2013-10-27T15:23:36Z","abstract_excerpt":"In a projective plane $\\Pi_q$ of order $q$, a non-empty point set ${\\cal S}_t$ is a $t$-semiarc if the number of tangent lines to ${\\cal S}_t$ at each of its points is $t$. If ${\\cal S}_t$ is a $t$-semiarc in $\\Pi_q$, $t<q$, then each line intersects ${\\cal S}_t$ in at most $q+1-t$ points. Dover proved that semiovals (semiarcs with $t=1$) containing $q$ collinear points exist in $\\Pi_q$ only if $q<3$. We show that if $t>1$, then $t$-semiarcs with $q+1-t$ collinear points exist only if $t\\geq \\sqrt{q-1}$. In $\\mathrm{PG}(2,q)$ we prove the lower bound $t\\geq(q-1)/2$, with equality only if ${\\ca"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1310.7204","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"}