{"paper":{"title":"On the use of Klein quadric for geometric incidence problems in two dimensions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.MG","authors_text":"J. M. Selig, Misha Rudnev","submitted_at":"2014-12-09T10:54:42Z","abstract_excerpt":"We discuss a unified approach to a class of geometric combinatorics incidence problems in $2D$, of the Erd\\\"os distance type. The goal is obtaining the second moment estimate, that is given a finite point set $S$ and a function $f$ on $S\\times S$, an upper bound on the number of solutions of $$\n  f(p,p') = f(q,q')\\neq 0,\\qquad (p,p',q,q')\\in S\\times S\\times S\\times S. \\qquad(*) $$ E.g., $f$ is the Euclidean distance in the plane, sphere, or a sheet of the two-sheeted hyperboloid.\n  Our tool is the Guth-Katz incidence theorem for lines in $\\mathbb{RP}^3$, but we focus on how the original $2D$ p"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1412.2909","kind":"arxiv","version":2},"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"}