{"paper":{"title":"Randomly piercing algebraic sets","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.NT","authors_text":"Daniel Altman, Nathan Tung","submitted_at":"2026-06-18T01:01:00Z","abstract_excerpt":"We show, for example, that if one samples \\[\\frac{\\log p}{2\\log(1+(p-1)^{-1})} \\cdot n^2(1 + o_{n\\to \\infty}(1))\\] points in $\\mathbb{F}_p^n$ at random then asymptotically almost surely this set intersects every quadratic hypersurface. We furthermore show that this is tight in that sampling $o_{n\\to\\infty}(n^2)$ fewer points almost surely fails to intersect some quadratic hypersurface.\n  Our main result is a sharp threshold for the following problem: how many points in $\\mathbb{F}_p^n$ does one need to randomly sample to almost surely intersect every algebraic set defined by at most $s$ polyno"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.19677","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2606.19677/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}