{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2013:F4XHRFEMBKVIZXEKL7ASTVM7DO","short_pith_number":"pith:F4XHRFEM","canonical_record":{"source":{"id":"1307.2490","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2013-07-09T15:21:14Z","cross_cats_sorted":[],"title_canon_sha256":"61a3c69838ce11343033de6ef25c77619fff8d71b68e94fb121e33b526bd69cd","abstract_canon_sha256":"3a539afc39825753ddb8eca5a1242f11900be0fe0ec91919c85d4ebf35d885ee"},"schema_version":"1.0"},"canonical_sha256":"2f2e78948c0aaa8cdc8a5fc129d59f1b8a612acdf0ff63e37ea8c2c446a5987b","source":{"kind":"arxiv","id":"1307.2490","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1307.2490","created_at":"2026-05-18T03:18:52Z"},{"alias_kind":"arxiv_version","alias_value":"1307.2490v1","created_at":"2026-05-18T03:18:52Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1307.2490","created_at":"2026-05-18T03:18:52Z"},{"alias_kind":"pith_short_12","alias_value":"F4XHRFEMBKVI","created_at":"2026-05-18T12:27:43Z"},{"alias_kind":"pith_short_16","alias_value":"F4XHRFEMBKVIZXEK","created_at":"2026-05-18T12:27:43Z"},{"alias_kind":"pith_short_8","alias_value":"F4XHRFEM","created_at":"2026-05-18T12:27:43Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2013:F4XHRFEMBKVIZXEKL7ASTVM7DO","target":"record","payload":{"canonical_record":{"source":{"id":"1307.2490","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2013-07-09T15:21:14Z","cross_cats_sorted":[],"title_canon_sha256":"61a3c69838ce11343033de6ef25c77619fff8d71b68e94fb121e33b526bd69cd","abstract_canon_sha256":"3a539afc39825753ddb8eca5a1242f11900be0fe0ec91919c85d4ebf35d885ee"},"schema_version":"1.0"},"canonical_sha256":"2f2e78948c0aaa8cdc8a5fc129d59f1b8a612acdf0ff63e37ea8c2c446a5987b","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:18:52.366137Z","signature_b64":"tBN72LaoDiZflJBXNvsgruFZ6iqk+joWW3IXSwkkv/p4RVlvUuF9XghmZhp32XVb37HgmIogpzFXgNJvxib2AA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"2f2e78948c0aaa8cdc8a5fc129d59f1b8a612acdf0ff63e37ea8c2c446a5987b","last_reissued_at":"2026-05-18T03:18:52.365578Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:18:52.365578Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1307.2490","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T03:18:52Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"SO3tNYhONx56QclSmP8wZjopsT3Mfp/zbvXU+H5kryCJARZGly1f20HdAYVisymTXusqAzFNgT6Qjm2M6BpIDg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-20T22:10:00.049935Z"},"content_sha256":"71baf101d00bc55847b7e7e9f394adbd1bfe3953ab53fe3394e229322d22bb2f","schema_version":"1.0","event_id":"sha256:71baf101d00bc55847b7e7e9f394adbd1bfe3953ab53fe3394e229322d22bb2f"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2013:F4XHRFEMBKVIZXEKL7ASTVM7DO","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"On the typical rank of real polynomials (or symmetric tensors) with a fixed border rank","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AG","authors_text":"Edoardo Ballico","submitted_at":"2013-07-09T15:21:14Z","abstract_excerpt":"Let $\\sigma_b(X_{m,d}(\\mathbb {C}))(\\mathbb {R})$, $b(m+1) < \\binom{m+d}{m}$, denote the set of all degree $d$ real homogeneous polynomials in $m+1$ variables (i.e. real symmetric tensors of format $(m+1)\\times ... \\times (m+1)$, $d$ times) which have border rank $b$ over $\\mathbb {C}$. It has a partition into manifolds of real dimension $\\le b(m+1)-1$ in which the real rank is constant. A typical rank of $\\sigma_b(X_{m,d}(\\mathbb {C}))(\\mathbb {R})$ is a rank associated to an open part of dimension $b(m+1)-1$. Here we classify all typical ranks when $b\\le 7$ and $d, m$ are not too small. For "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1307.2490","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":""},"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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T03:18:52Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"C3MX04nK2xwgf8oFbBK3xKWqQQrZQPK95gME64EwS8KZcXnLmd1EiBn9faUwADqna0yYfMCmKvaWhRsoMNAvDA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-20T22:10:00.050281Z"},"content_sha256":"2b9edc65f05f07ca5f3539409c212c551a035e738930eecbe03fdf26d03af14a","schema_version":"1.0","event_id":"sha256:2b9edc65f05f07ca5f3539409c212c551a035e738930eecbe03fdf26d03af14a"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/F4XHRFEMBKVIZXEKL7ASTVM7DO/bundle.json","state_url":"https://pith.science/pith/F4XHRFEMBKVIZXEKL7ASTVM7DO/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/F4XHRFEMBKVIZXEKL7ASTVM7DO/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-06-20T22:10:00Z","links":{"resolver":"https://pith.science/pith/F4XHRFEMBKVIZXEKL7ASTVM7DO","bundle":"https://pith.science/pith/F4XHRFEMBKVIZXEKL7ASTVM7DO/bundle.json","state":"https://pith.science/pith/F4XHRFEMBKVIZXEKL7ASTVM7DO/state.json","well_known_bundle":"https://pith.science/.well-known/pith/F4XHRFEMBKVIZXEKL7ASTVM7DO/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2013:F4XHRFEMBKVIZXEKL7ASTVM7DO","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":"3a539afc39825753ddb8eca5a1242f11900be0fe0ec91919c85d4ebf35d885ee","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2013-07-09T15:21:14Z","title_canon_sha256":"61a3c69838ce11343033de6ef25c77619fff8d71b68e94fb121e33b526bd69cd"},"schema_version":"1.0","source":{"id":"1307.2490","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1307.2490","created_at":"2026-05-18T03:18:52Z"},{"alias_kind":"arxiv_version","alias_value":"1307.2490v1","created_at":"2026-05-18T03:18:52Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1307.2490","created_at":"2026-05-18T03:18:52Z"},{"alias_kind":"pith_short_12","alias_value":"F4XHRFEMBKVI","created_at":"2026-05-18T12:27:43Z"},{"alias_kind":"pith_short_16","alias_value":"F4XHRFEMBKVIZXEK","created_at":"2026-05-18T12:27:43Z"},{"alias_kind":"pith_short_8","alias_value":"F4XHRFEM","created_at":"2026-05-18T12:27:43Z"}],"graph_snapshots":[{"event_id":"sha256:2b9edc65f05f07ca5f3539409c212c551a035e738930eecbe03fdf26d03af14a","target":"graph","created_at":"2026-05-18T03:18:52Z","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":"Let $\\sigma_b(X_{m,d}(\\mathbb {C}))(\\mathbb {R})$, $b(m+1) < \\binom{m+d}{m}$, denote the set of all degree $d$ real homogeneous polynomials in $m+1$ variables (i.e. real symmetric tensors of format $(m+1)\\times ... \\times (m+1)$, $d$ times) which have border rank $b$ over $\\mathbb {C}$. It has a partition into manifolds of real dimension $\\le b(m+1)-1$ in which the real rank is constant. A typical rank of $\\sigma_b(X_{m,d}(\\mathbb {C}))(\\mathbb {R})$ is a rank associated to an open part of dimension $b(m+1)-1$. Here we classify all typical ranks when $b\\le 7$ and $d, m$ are not too small. For ","authors_text":"Edoardo Ballico","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2013-07-09T15:21:14Z","title":"On the typical rank of real polynomials (or symmetric tensors) with a fixed border rank"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1307.2490","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":"sha256:71baf101d00bc55847b7e7e9f394adbd1bfe3953ab53fe3394e229322d22bb2f","target":"record","created_at":"2026-05-18T03:18:52Z","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":"3a539afc39825753ddb8eca5a1242f11900be0fe0ec91919c85d4ebf35d885ee","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2013-07-09T15:21:14Z","title_canon_sha256":"61a3c69838ce11343033de6ef25c77619fff8d71b68e94fb121e33b526bd69cd"},"schema_version":"1.0","source":{"id":"1307.2490","kind":"arxiv","version":1}},"canonical_sha256":"2f2e78948c0aaa8cdc8a5fc129d59f1b8a612acdf0ff63e37ea8c2c446a5987b","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"2f2e78948c0aaa8cdc8a5fc129d59f1b8a612acdf0ff63e37ea8c2c446a5987b","first_computed_at":"2026-05-18T03:18:52.365578Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:18:52.365578Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"tBN72LaoDiZflJBXNvsgruFZ6iqk+joWW3IXSwkkv/p4RVlvUuF9XghmZhp32XVb37HgmIogpzFXgNJvxib2AA==","signature_status":"signed_v1","signed_at":"2026-05-18T03:18:52.366137Z","signed_message":"canonical_sha256_bytes"},"source_id":"1307.2490","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:71baf101d00bc55847b7e7e9f394adbd1bfe3953ab53fe3394e229322d22bb2f","sha256:2b9edc65f05f07ca5f3539409c212c551a035e738930eecbe03fdf26d03af14a"],"state_sha256":"5aa304e3e3ef2f3c4a20c0b9115a2709d8b54cdd98ad1d772b92c5b861ef4f75"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"uKovZLf+5F/7+I5D7hvqUpx3ccTugKSSfJoc7l1uRo47wVrUWKoMUmcQQUh6B7DLlvmu2gS3ANlzORhShoWrAg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-20T22:10:00.052232Z","bundle_sha256":"4c5efab1ce1090533b382fbf113385b1194eaec41870e1b0ed06380f83821778"}}