{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2018:GGP477CCFV64YCVBCYTOQAKAIT","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":"21d60336cb0305032cefe92f6fc4110fe301857165f1cd5b982c0a6cc6388781","cross_cats_sorted":["math.NT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-09-28T09:27:41Z","title_canon_sha256":"eb38779bdb30b62195ecad579244751e194b32f23903f0909a746e0a2162e622"},"schema_version":"1.0","source":{"id":"1809.10931","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1809.10931","created_at":"2026-05-18T00:04:34Z"},{"alias_kind":"arxiv_version","alias_value":"1809.10931v1","created_at":"2026-05-18T00:04:34Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1809.10931","created_at":"2026-05-18T00:04:34Z"},{"alias_kind":"pith_short_12","alias_value":"GGP477CCFV64","created_at":"2026-05-18T12:32:25Z"},{"alias_kind":"pith_short_16","alias_value":"GGP477CCFV64YCVB","created_at":"2026-05-18T12:32:25Z"},{"alias_kind":"pith_short_8","alias_value":"GGP477CC","created_at":"2026-05-18T12:32:25Z"}],"graph_snapshots":[{"event_id":"sha256:e78ba63d8cff95b64224b2945da1d2328d9811a6dff8507a33a67f26e4ed4cac","target":"graph","created_at":"2026-05-18T00:04:34Z","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":"A tensor defined over a finite field $\\mathbb{F}$ has low analytic rank if the distribution of its values differs significantly from the uniform distribution. An order $d$ tensor has partition rank 1 if it can be written as a product of two tensors of order less than $d$, and it has partition rank at most $k$ if it can be written as a sum of $k$ tensors of partition rank 1. In this paper, we prove that if the analytic rank of an order $d$ tensor is at most $r$, then its partition rank is at most $f(r,d,|\\mathbb{F}|)$. Previously, this was known with $f$ being an Ackermann-type function in $r$ ","authors_text":"Oliver Janzer","cross_cats":["math.NT"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-09-28T09:27:41Z","title":"Low analytic rank implies low partition rank for tensors"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1809.10931","kind":"arxiv","version":1},"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":"sha256:2b5d08b6a1bf8ce13441baf576bfe3530da592c0583b1c38347e196588ec5c0b","target":"record","created_at":"2026-05-18T00:04:34Z","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":"21d60336cb0305032cefe92f6fc4110fe301857165f1cd5b982c0a6cc6388781","cross_cats_sorted":["math.NT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-09-28T09:27:41Z","title_canon_sha256":"eb38779bdb30b62195ecad579244751e194b32f23903f0909a746e0a2162e622"},"schema_version":"1.0","source":{"id":"1809.10931","kind":"arxiv","version":1}},"canonical_sha256":"319fcffc422d7dcc0aa11626e8014044d045ad2a6a31dc0f44bd1254588dc08e","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"319fcffc422d7dcc0aa11626e8014044d045ad2a6a31dc0f44bd1254588dc08e","first_computed_at":"2026-05-18T00:04:34.575250Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:04:34.575250Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"mp0wtdleMspOLvb/IsPfbY+OnV26VA5VUqjqyX1hU0JlwrQ8BcbbewyFRAUKY7sF9Ury4vdcUMMsayJjbJKeCQ==","signature_status":"signed_v1","signed_at":"2026-05-18T00:04:34.575854Z","signed_message":"canonical_sha256_bytes"},"source_id":"1809.10931","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:2b5d08b6a1bf8ce13441baf576bfe3530da592c0583b1c38347e196588ec5c0b","sha256:e78ba63d8cff95b64224b2945da1d2328d9811a6dff8507a33a67f26e4ed4cac"],"state_sha256":"919e006dbf108e7c90073cdf30560e5797b06b90c93b2ddb1963efe535d03dec"}