{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2014:MY3UHCFC34GR3ZD65P4W3MJRYI","short_pith_number":"pith:MY3UHCFC","schema_version":"1.0","canonical_sha256":"66374388a2df0d1de47eebf96db131c237f3dd84373558338bd6516f80fa52cc","source":{"kind":"arxiv","id":"1408.4711","version":2},"attestation_state":"computed","paper":{"title":"Minimizing Cubic and Homogeneous Polynomials over Integers in the Plane","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.OC","authors_text":"Alberto Del Pia, Kevin Zemmer, Robert Hildebrand, Robert Weismantel","submitted_at":"2014-08-20T16:15:03Z","abstract_excerpt":"We complete the complexity classification by degree of minimizing a polynomial over the integer points in a polyhedron in $\\mathbb{R}^2$. Previous work shows that optimizing a quadratic polynomial over the integer points in a polyhedral region in $\\mathbb{R}^2$ can be done in polynomial time, while optimizing a quartic polynomial in the same type of region is NP-hard. We close the gap by showing that this problem can be solved in polynomial time for cubic polynomials.\n  Furthermore, we show that the problem of minimizing a homogeneous polynomial of any fixed degree over the integer points in a"},"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":"1408.4711","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OC","submitted_at":"2014-08-20T16:15:03Z","cross_cats_sorted":[],"title_canon_sha256":"004d15245cfa44a1f3360de98e56ec66aff371bc7b1ed76aa5a262015841c3f5","abstract_canon_sha256":"bfd086545578317e079f3098435a84ed6b6e286586d35026c0c3011805cec6a4"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:16:51.198589Z","signature_b64":"5vow7+wET8Sa5XQ1rrtZAYl3IZOx6cn1IC/x3SObZ7uEgojFCVnXjuwezJtP9gwn/2uNxaeYJxXAvR1czPrmAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"66374388a2df0d1de47eebf96db131c237f3dd84373558338bd6516f80fa52cc","last_reissued_at":"2026-05-18T02:16:51.197967Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:16:51.197967Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Minimizing Cubic and Homogeneous Polynomials over Integers in the Plane","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.OC","authors_text":"Alberto Del Pia, Kevin Zemmer, Robert Hildebrand, Robert Weismantel","submitted_at":"2014-08-20T16:15:03Z","abstract_excerpt":"We complete the complexity classification by degree of minimizing a polynomial over the integer points in a polyhedron in $\\mathbb{R}^2$. Previous work shows that optimizing a quadratic polynomial over the integer points in a polyhedral region in $\\mathbb{R}^2$ can be done in polynomial time, while optimizing a quartic polynomial in the same type of region is NP-hard. We close the gap by showing that this problem can be solved in polynomial time for cubic polynomials.\n  Furthermore, we show that the problem of minimizing a homogeneous polynomial of any fixed degree over the integer points in a"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1408.4711","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":"1408.4711","created_at":"2026-05-18T02:16:51.198062+00:00"},{"alias_kind":"arxiv_version","alias_value":"1408.4711v2","created_at":"2026-05-18T02:16:51.198062+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1408.4711","created_at":"2026-05-18T02:16:51.198062+00:00"},{"alias_kind":"pith_short_12","alias_value":"MY3UHCFC34GR","created_at":"2026-05-18T12:28:38.356838+00:00"},{"alias_kind":"pith_short_16","alias_value":"MY3UHCFC34GR3ZD6","created_at":"2026-05-18T12:28:38.356838+00:00"},{"alias_kind":"pith_short_8","alias_value":"MY3UHCFC","created_at":"2026-05-18T12:28:38.356838+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/MY3UHCFC34GR3ZD65P4W3MJRYI","json":"https://pith.science/pith/MY3UHCFC34GR3ZD65P4W3MJRYI.json","graph_json":"https://pith.science/api/pith-number/MY3UHCFC34GR3ZD65P4W3MJRYI/graph.json","events_json":"https://pith.science/api/pith-number/MY3UHCFC34GR3ZD65P4W3MJRYI/events.json","paper":"https://pith.science/paper/MY3UHCFC"},"agent_actions":{"view_html":"https://pith.science/pith/MY3UHCFC34GR3ZD65P4W3MJRYI","download_json":"https://pith.science/pith/MY3UHCFC34GR3ZD65P4W3MJRYI.json","view_paper":"https://pith.science/paper/MY3UHCFC","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1408.4711&json=true","fetch_graph":"https://pith.science/api/pith-number/MY3UHCFC34GR3ZD65P4W3MJRYI/graph.json","fetch_events":"https://pith.science/api/pith-number/MY3UHCFC34GR3ZD65P4W3MJRYI/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/MY3UHCFC34GR3ZD65P4W3MJRYI/action/timestamp_anchor","attest_storage":"https://pith.science/pith/MY3UHCFC34GR3ZD65P4W3MJRYI/action/storage_attestation","attest_author":"https://pith.science/pith/MY3UHCFC34GR3ZD65P4W3MJRYI/action/author_attestation","sign_citation":"https://pith.science/pith/MY3UHCFC34GR3ZD65P4W3MJRYI/action/citation_signature","submit_replication":"https://pith.science/pith/MY3UHCFC34GR3ZD65P4W3MJRYI/action/replication_record"}},"created_at":"2026-05-18T02:16:51.198062+00:00","updated_at":"2026-05-18T02:16:51.198062+00:00"}