{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2014:L7ROGDXEBGEOWQEYOVQJ74PCEH","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"9a34f5dc473e0958de6aee9c5511f25daf86a4c5b921b58e8ec2b5be8ccaabc5","cross_cats_sorted":["math.NT","math.OC"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2014-09-18T11:09:32Z","title_canon_sha256":"8dd9c513310737284b98ed7e35c0704979b0ff39f8df4198c3ce1429ca4ea2ec"},"schema_version":"1.0","source":{"id":"1409.5259","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1409.5259","created_at":"2026-05-18T01:36:22Z"},{"alias_kind":"arxiv_version","alias_value":"1409.5259v2","created_at":"2026-05-18T01:36:22Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1409.5259","created_at":"2026-05-18T01:36:22Z"},{"alias_kind":"pith_short_12","alias_value":"L7ROGDXEBGEO","created_at":"2026-05-18T12:28:35Z"},{"alias_kind":"pith_short_16","alias_value":"L7ROGDXEBGEOWQEY","created_at":"2026-05-18T12:28:35Z"},{"alias_kind":"pith_short_8","alias_value":"L7ROGDXE","created_at":"2026-05-18T12:28:35Z"}],"graph_snapshots":[{"event_id":"sha256:8cdfe5721bbc452a1680b9d292b310d79552668bace284a445aa06be6f538e8e","target":"graph","created_at":"2026-05-18T01:36:22Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"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. ","authors_text":"Iskander Aliev, Jesus A. De Loera, Quentin Louveaux","cross_cats":["math.NT","math.OC"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2014-09-18T11:09:32Z","title":"Parametric Polyhedra with at least $k$ Lattice Points: Their Semigroup Structure and the k-Frobenius Problem"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1409.5259","kind":"arxiv","version":2},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:f7ea936b9556c562bf12e36b32230750d98bb959c845134eab3a17d01497d13e","target":"record","created_at":"2026-05-18T01:36:22Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"9a34f5dc473e0958de6aee9c5511f25daf86a4c5b921b58e8ec2b5be8ccaabc5","cross_cats_sorted":["math.NT","math.OC"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2014-09-18T11:09:32Z","title_canon_sha256":"8dd9c513310737284b98ed7e35c0704979b0ff39f8df4198c3ce1429ca4ea2ec"},"schema_version":"1.0","source":{"id":"1409.5259","kind":"arxiv","version":2}},"canonical_sha256":"5fe2e30ee40988eb409875609ff1e221c108a9d7cdab014d769bd031432c7867","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"5fe2e30ee40988eb409875609ff1e221c108a9d7cdab014d769bd031432c7867","first_computed_at":"2026-05-18T01:36:22.260650Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:36:22.260650Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"x9KoXaXZM2i0dXNnO4p6670uVxxl2Tuxl6I8RpCFSWRZnrdVi3F8J1q5xS0YDmL+v84nDznyjQRH8jIUtMhJBQ==","signature_status":"signed_v1","signed_at":"2026-05-18T01:36:22.261121Z","signed_message":"canonical_sha256_bytes"},"source_id":"1409.5259","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:f7ea936b9556c562bf12e36b32230750d98bb959c845134eab3a17d01497d13e","sha256:8cdfe5721bbc452a1680b9d292b310d79552668bace284a445aa06be6f538e8e"],"state_sha256":"9bf6695220e3db70e395f1de005ad6e8957da9a0e2ed1765b44f85e7e455a714"}