{"paper":{"title":"On inscribed trapezoids and affinely 3-regular maps","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.GT","authors_text":"Florian Frick, Michael Harrison","submitted_at":"2021-09-13T14:04:07Z","abstract_excerpt":"We show that any embedding $\\mathbb{R}^d \\to \\mathbb{R}^{2d+2^{\\gamma(d)}-1}$ inscribes a trapezoid or maps three points to a line, where $2^{\\gamma(d)}$ is the smallest power of $2$ satisfying $2^{\\gamma(d)} \\geq \\rho(d)$, and $\\rho(d)$ denotes the Hurwitz--Radon function. The proof is elementary and includes a novel application of nonsingular bilinear maps. As an application, we recover recent results on the nonexistence of affinely $3$-regular maps, for infinitely many dimensions $d$, without resorting to sophisticated algebraic techniques."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2109.05985","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2109.05985/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"}