{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2014:L7ROGDXEBGEOWQEYOVQJ74PCEH","short_pith_number":"pith:L7ROGDXE","schema_version":"1.0","canonical_sha256":"5fe2e30ee40988eb409875609ff1e221c108a9d7cdab014d769bd031432c7867","source":{"kind":"arxiv","id":"1409.5259","version":2},"attestation_state":"computed","paper":{"title":"Parametric Polyhedra with at least $k$ Lattice Points: Their Semigroup Structure and the k-Frobenius Problem","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.NT","math.OC"],"primary_cat":"math.CO","authors_text":"Iskander Aliev, Jesus A. De Loera, Quentin Louveaux","submitted_at":"2014-09-18T11:09:32Z","abstract_excerpt":"Given an integral $d \\times n$ matrix $A$, the well-studied affine semigroup $\\mbox{ Sg} (A)=\\{ b : Ax=b, \\ x \\in {\\mathbb Z}^n, x \\geq 0\\}$ can be stratified by the number of lattice points inside the parametric polyhedra $P_A(b)=\\{x: Ax=b, x\\geq0\\}$. Such families of parametric polyhedra appear in many areas of combinatorics, convex geometry, algebra and number theory. The key themes of this paper are: (1) A structure theory that characterizes precisely the subset $\\mbox{ Sg}_{\\geq k}(A)$ of all vectors $b \\in \\mbox{ Sg}(A)$ such that $P_A(b) \\cap {\\mathbb Z}^n $ has at least $k$ solutions. "},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1409.5259","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2014-09-18T11:09:32Z","cross_cats_sorted":["math.NT","math.OC"],"title_canon_sha256":"8dd9c513310737284b98ed7e35c0704979b0ff39f8df4198c3ce1429ca4ea2ec","abstract_canon_sha256":"9a34f5dc473e0958de6aee9c5511f25daf86a4c5b921b58e8ec2b5be8ccaabc5"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:36:22.261121Z","signature_b64":"x9KoXaXZM2i0dXNnO4p6670uVxxl2Tuxl6I8RpCFSWRZnrdVi3F8J1q5xS0YDmL+v84nDznyjQRH8jIUtMhJBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"5fe2e30ee40988eb409875609ff1e221c108a9d7cdab014d769bd031432c7867","last_reissued_at":"2026-05-18T01:36:22.260650Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:36:22.260650Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Parametric Polyhedra with at least $k$ Lattice Points: Their Semigroup Structure and the k-Frobenius Problem","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.NT","math.OC"],"primary_cat":"math.CO","authors_text":"Iskander Aliev, Jesus A. De Loera, Quentin Louveaux","submitted_at":"2014-09-18T11:09:32Z","abstract_excerpt":"Given an integral $d \\times n$ matrix $A$, the well-studied affine semigroup $\\mbox{ Sg} (A)=\\{ b : Ax=b, \\ x \\in {\\mathbb Z}^n, x \\geq 0\\}$ can be stratified by the number of lattice points inside the parametric polyhedra $P_A(b)=\\{x: Ax=b, x\\geq0\\}$. Such families of parametric polyhedra appear in many areas of combinatorics, convex geometry, algebra and number theory. The key themes of this paper are: (1) A structure theory that characterizes precisely the subset $\\mbox{ Sg}_{\\geq k}(A)$ of all vectors $b \\in \\mbox{ Sg}(A)$ such that $P_A(b) \\cap {\\mathbb Z}^n $ has at least $k$ solutions. "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1409.5259","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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1409.5259","created_at":"2026-05-18T01:36:22.260723+00:00"},{"alias_kind":"arxiv_version","alias_value":"1409.5259v2","created_at":"2026-05-18T01:36:22.260723+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1409.5259","created_at":"2026-05-18T01:36:22.260723+00:00"},{"alias_kind":"pith_short_12","alias_value":"L7ROGDXEBGEO","created_at":"2026-05-18T12:28:35.611951+00:00"},{"alias_kind":"pith_short_16","alias_value":"L7ROGDXEBGEOWQEY","created_at":"2026-05-18T12:28:35.611951+00:00"},{"alias_kind":"pith_short_8","alias_value":"L7ROGDXE","created_at":"2026-05-18T12:28:35.611951+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/L7ROGDXEBGEOWQEYOVQJ74PCEH","json":"https://pith.science/pith/L7ROGDXEBGEOWQEYOVQJ74PCEH.json","graph_json":"https://pith.science/api/pith-number/L7ROGDXEBGEOWQEYOVQJ74PCEH/graph.json","events_json":"https://pith.science/api/pith-number/L7ROGDXEBGEOWQEYOVQJ74PCEH/events.json","paper":"https://pith.science/paper/L7ROGDXE"},"agent_actions":{"view_html":"https://pith.science/pith/L7ROGDXEBGEOWQEYOVQJ74PCEH","download_json":"https://pith.science/pith/L7ROGDXEBGEOWQEYOVQJ74PCEH.json","view_paper":"https://pith.science/paper/L7ROGDXE","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1409.5259&json=true","fetch_graph":"https://pith.science/api/pith-number/L7ROGDXEBGEOWQEYOVQJ74PCEH/graph.json","fetch_events":"https://pith.science/api/pith-number/L7ROGDXEBGEOWQEYOVQJ74PCEH/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/L7ROGDXEBGEOWQEYOVQJ74PCEH/action/timestamp_anchor","attest_storage":"https://pith.science/pith/L7ROGDXEBGEOWQEYOVQJ74PCEH/action/storage_attestation","attest_author":"https://pith.science/pith/L7ROGDXEBGEOWQEYOVQJ74PCEH/action/author_attestation","sign_citation":"https://pith.science/pith/L7ROGDXEBGEOWQEYOVQJ74PCEH/action/citation_signature","submit_replication":"https://pith.science/pith/L7ROGDXEBGEOWQEYOVQJ74PCEH/action/replication_record"}},"created_at":"2026-05-18T01:36:22.260723+00:00","updated_at":"2026-05-18T01:36:22.260723+00:00"}