{"paper":{"title":"Basic geometry of the affine group over Z","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DS","authors_text":"Daniele Mundici","submitted_at":"2019-02-03T21:35:05Z","abstract_excerpt":"The subject matter of this paper is the geometry of the affine group over the integers,\n  $\\mathsf{GL}(n,\\mathbb{Z})\\ltimes \\mathbb{Z}^n$. Turing-computable complete $\\mathsf{GL}(n,\\mathbb{Z})\\ltimes \\mathbb{Z}^n$-orbit invariants are constructed for angles, segments, triangles and ellipses. In rational affine $\\mathsf{GL}(n,\\mathbb Q)\\ltimes \\mathbb Q^n$-geometry, ellipses are classified by the Clifford--Hasse--Witt invariant, via the Hasse-Minkowski theorem. We classify ellipses in $\\mathsf{GL}(n,\\mathbb{Z})\\ltimes \\mathbb{Z}^n$-geometry combining results by Apollonius of Perga and Pappus of"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1902.00971","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"}