{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2019:SDYPE6IGFOF5J6WCHQCE46AJ2P","short_pith_number":"pith:SDYPE6IG","schema_version":"1.0","canonical_sha256":"90f0f279062b8bd4fac23c044e7809d3c0f2843ebbbf583e6ddbddd437b45d12","source":{"kind":"arxiv","id":"1910.07271","version":1},"attestation_state":"computed","paper":{"title":"Representation of Polytopes as Polynomial Zonotopes","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Matthias Althoff, Niklas Kochdumper","submitted_at":"2019-10-16T10:40:54Z","abstract_excerpt":"We prove that each bounded polytope can be represented as a polynomial zonotope, which we refer to as the Z-representation of polytopes. Previous representations are the vertex representation (V-representation) and the halfspace representation (H-representation). Depending on the polytope, the Z-representation can be more compact than the V-representation and the H-representation. In addition, the Z-representation enables the computation of linear maps, Minkowski addition, and convex hull with a computational complexity that is polynomial in the representation size. The usefulness of the new r"},"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":"1910.07271","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2019-10-16T10:40:54Z","cross_cats_sorted":[],"title_canon_sha256":"cda3fda281f733a315e728f971134cabb832a65f0fb504d8b39c5eb5aa382330","abstract_canon_sha256":"bb5fa854c1ac7f1748a8b9c202f515fff6fc059602d6f59c1b6a3225bdab0f24"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T00:12:34.370682Z","signature_b64":"I+qHcUUnx1MZ/7lx/dicNhrYbzEW+CjTQqV7XkqRs974Wo1RQdyhF8QClGlN8fCCzWGfLNEO+2nLPFE/oB5LBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"90f0f279062b8bd4fac23c044e7809d3c0f2843ebbbf583e6ddbddd437b45d12","last_reissued_at":"2026-07-05T00:12:34.370351Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T00:12:34.370351Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Representation of Polytopes as Polynomial Zonotopes","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Matthias Althoff, Niklas Kochdumper","submitted_at":"2019-10-16T10:40:54Z","abstract_excerpt":"We prove that each bounded polytope can be represented as a polynomial zonotope, which we refer to as the Z-representation of polytopes. Previous representations are the vertex representation (V-representation) and the halfspace representation (H-representation). Depending on the polytope, the Z-representation can be more compact than the V-representation and the H-representation. In addition, the Z-representation enables the computation of linear maps, Minkowski addition, and convex hull with a computational complexity that is polynomial in the representation size. The usefulness of the new r"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1910.07271","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/1910.07271/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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1910.07271","created_at":"2026-07-05T00:12:34.370408+00:00"},{"alias_kind":"arxiv_version","alias_value":"1910.07271v1","created_at":"2026-07-05T00:12:34.370408+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1910.07271","created_at":"2026-07-05T00:12:34.370408+00:00"},{"alias_kind":"pith_short_12","alias_value":"SDYPE6IGFOF5","created_at":"2026-07-05T00:12:34.370408+00:00"},{"alias_kind":"pith_short_16","alias_value":"SDYPE6IGFOF5J6WC","created_at":"2026-07-05T00:12:34.370408+00:00"},{"alias_kind":"pith_short_8","alias_value":"SDYPE6IG","created_at":"2026-07-05T00:12:34.370408+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":1,"internal_anchor_count":1,"sample":[{"citing_arxiv_id":"2502.01360","citing_title":"A Quotient Homology Theory of Representation in Neural Networks","ref_index":26,"is_internal_anchor":true}]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/SDYPE6IGFOF5J6WCHQCE46AJ2P","json":"https://pith.science/pith/SDYPE6IGFOF5J6WCHQCE46AJ2P.json","graph_json":"https://pith.science/api/pith-number/SDYPE6IGFOF5J6WCHQCE46AJ2P/graph.json","events_json":"https://pith.science/api/pith-number/SDYPE6IGFOF5J6WCHQCE46AJ2P/events.json","paper":"https://pith.science/paper/SDYPE6IG"},"agent_actions":{"view_html":"https://pith.science/pith/SDYPE6IGFOF5J6WCHQCE46AJ2P","download_json":"https://pith.science/pith/SDYPE6IGFOF5J6WCHQCE46AJ2P.json","view_paper":"https://pith.science/paper/SDYPE6IG","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1910.07271&json=true","fetch_graph":"https://pith.science/api/pith-number/SDYPE6IGFOF5J6WCHQCE46AJ2P/graph.json","fetch_events":"https://pith.science/api/pith-number/SDYPE6IGFOF5J6WCHQCE46AJ2P/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/SDYPE6IGFOF5J6WCHQCE46AJ2P/action/timestamp_anchor","attest_storage":"https://pith.science/pith/SDYPE6IGFOF5J6WCHQCE46AJ2P/action/storage_attestation","attest_author":"https://pith.science/pith/SDYPE6IGFOF5J6WCHQCE46AJ2P/action/author_attestation","sign_citation":"https://pith.science/pith/SDYPE6IGFOF5J6WCHQCE46AJ2P/action/citation_signature","submit_replication":"https://pith.science/pith/SDYPE6IGFOF5J6WCHQCE46AJ2P/action/replication_record"}},"created_at":"2026-07-05T00:12:34.370408+00:00","updated_at":"2026-07-05T00:12:34.370408+00:00"}