{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2012:4MSX2TUXERB3OCKOMPNZGOOQGY","short_pith_number":"pith:4MSX2TUX","canonical_record":{"source":{"id":"1211.3569","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2012-11-15T10:46:47Z","cross_cats_sorted":["math.AC"],"title_canon_sha256":"9b4f270199d29bef908aa93555384ca4602ce83d1ab3f95b85d912214026d0a6","abstract_canon_sha256":"987ad78fa652a77982cf5a04d940c16349c462cf49272c1e31f36238000ceaf8"},"schema_version":"1.0"},"canonical_sha256":"e3257d4e972443b7094e63db9339d03618dd53246ea103afd6e6ba94f2f4d96f","source":{"kind":"arxiv","id":"1211.3569","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1211.3569","created_at":"2026-05-18T03:40:39Z"},{"alias_kind":"arxiv_version","alias_value":"1211.3569v1","created_at":"2026-05-18T03:40:39Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1211.3569","created_at":"2026-05-18T03:40:39Z"},{"alias_kind":"pith_short_12","alias_value":"4MSX2TUXERB3","created_at":"2026-05-18T12:26:53Z"},{"alias_kind":"pith_short_16","alias_value":"4MSX2TUXERB3OCKO","created_at":"2026-05-18T12:26:53Z"},{"alias_kind":"pith_short_8","alias_value":"4MSX2TUX","created_at":"2026-05-18T12:26:53Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2012:4MSX2TUXERB3OCKOMPNZGOOQGY","target":"record","payload":{"canonical_record":{"source":{"id":"1211.3569","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2012-11-15T10:46:47Z","cross_cats_sorted":["math.AC"],"title_canon_sha256":"9b4f270199d29bef908aa93555384ca4602ce83d1ab3f95b85d912214026d0a6","abstract_canon_sha256":"987ad78fa652a77982cf5a04d940c16349c462cf49272c1e31f36238000ceaf8"},"schema_version":"1.0"},"canonical_sha256":"e3257d4e972443b7094e63db9339d03618dd53246ea103afd6e6ba94f2f4d96f","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:40:39.398601Z","signature_b64":"MxFQ+b2vmmydKuKR+eQThsIdf+8iE5mDNaFL8MO/C+7+SvxqbAdJ07cyHdQtgNVgQPBrzm1SHxTBkgR99Nq3AA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"e3257d4e972443b7094e63db9339d03618dd53246ea103afd6e6ba94f2f4d96f","last_reissued_at":"2026-05-18T03:40:39.397918Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:40:39.397918Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1211.3569","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":"2026-05-18T03:40:39Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"XphcrnLoLFNIns8AmSJHZ2zbvrZuXAgpaz4CnHHf4o6y4L1tuGYM0toCIsRhdsI3VhJ0BJ+GcX92eIgVa3OVAw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-27T14:44:56.466113Z"},"content_sha256":"b4ba6ce9d34745332c0e010e7e8de0c32768a827bd35a75a5f03f472de7208cd","schema_version":"1.0","event_id":"sha256:b4ba6ce9d34745332c0e010e7e8de0c32768a827bd35a75a5f03f472de7208cd"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2012:4MSX2TUXERB3OCKOMPNZGOOQGY","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Low rank approximation of polynomials","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AC"],"primary_cat":"math.CO","authors_text":"Alexander Schrijver","submitted_at":"2012-11-15T10:46:47Z","abstract_excerpt":"Let $k\\leq n$. Each polynomial $p\\in\\oR[x_1,...,x_n]$ can be uniquely written as $p=\\sum_{\\mu}\\mu p_{\\mu}$, where $\\mu$ ranges over the set $M$ of all monomials in $\\oR[x_1,...,x_k]$ and where $p_{\\mu}\\in\\oR[x_{k+1},...,x_n]$. If $p$ is $d$-homogeneous and $\\varepsilon>0$, we say that $p$ is {\\em $\\varepsilon$-concentrated on the first $k$ variables} if $$\\sum_{\\mu\\in M\\atop\\deg(\\mu)<d}\\max_{x\\in\\oR^{n-k}\\atop\\|x\\|=1}p_{\\mu}(x)^2\\leq\\varepsilon\\|p\\|^2,$$ where $\\|p\\|$ is the Bombieri norm of $p$. We show that for each $d\\in\\oN$ and $\\varepsilon>0$ there exists $k_{d,\\varepsilon}$ such that for"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1211.3569","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":"2026-05-18T03:40:39Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"M5KV2TyJKwBYCqmyVuybJDiYtaxinwqpzYEJe4J5ehWRBBkG0X202Mj/0NtjCaYbwJkla/K3elAkeDeii1zgCA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-27T14:44:56.466469Z"},"content_sha256":"5dd0ac8f3fb6e1e701897d963563f09ba5ccb4c9fcf050d19585e394f5c9e7c4","schema_version":"1.0","event_id":"sha256:5dd0ac8f3fb6e1e701897d963563f09ba5ccb4c9fcf050d19585e394f5c9e7c4"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/4MSX2TUXERB3OCKOMPNZGOOQGY/bundle.json","state_url":"https://pith.science/pith/4MSX2TUXERB3OCKOMPNZGOOQGY/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/4MSX2TUXERB3OCKOMPNZGOOQGY/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":"2026-06-27T14:44:56Z","links":{"resolver":"https://pith.science/pith/4MSX2TUXERB3OCKOMPNZGOOQGY","bundle":"https://pith.science/pith/4MSX2TUXERB3OCKOMPNZGOOQGY/bundle.json","state":"https://pith.science/pith/4MSX2TUXERB3OCKOMPNZGOOQGY/state.json","well_known_bundle":"https://pith.science/.well-known/pith/4MSX2TUXERB3OCKOMPNZGOOQGY/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2012:4MSX2TUXERB3OCKOMPNZGOOQGY","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":"987ad78fa652a77982cf5a04d940c16349c462cf49272c1e31f36238000ceaf8","cross_cats_sorted":["math.AC"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2012-11-15T10:46:47Z","title_canon_sha256":"9b4f270199d29bef908aa93555384ca4602ce83d1ab3f95b85d912214026d0a6"},"schema_version":"1.0","source":{"id":"1211.3569","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1211.3569","created_at":"2026-05-18T03:40:39Z"},{"alias_kind":"arxiv_version","alias_value":"1211.3569v1","created_at":"2026-05-18T03:40:39Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1211.3569","created_at":"2026-05-18T03:40:39Z"},{"alias_kind":"pith_short_12","alias_value":"4MSX2TUXERB3","created_at":"2026-05-18T12:26:53Z"},{"alias_kind":"pith_short_16","alias_value":"4MSX2TUXERB3OCKO","created_at":"2026-05-18T12:26:53Z"},{"alias_kind":"pith_short_8","alias_value":"4MSX2TUX","created_at":"2026-05-18T12:26:53Z"}],"graph_snapshots":[{"event_id":"sha256:5dd0ac8f3fb6e1e701897d963563f09ba5ccb4c9fcf050d19585e394f5c9e7c4","target":"graph","created_at":"2026-05-18T03:40:39Z","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 $k\\leq n$. Each polynomial $p\\in\\oR[x_1,...,x_n]$ can be uniquely written as $p=\\sum_{\\mu}\\mu p_{\\mu}$, where $\\mu$ ranges over the set $M$ of all monomials in $\\oR[x_1,...,x_k]$ and where $p_{\\mu}\\in\\oR[x_{k+1},...,x_n]$. If $p$ is $d$-homogeneous and $\\varepsilon>0$, we say that $p$ is {\\em $\\varepsilon$-concentrated on the first $k$ variables} if $$\\sum_{\\mu\\in M\\atop\\deg(\\mu)<d}\\max_{x\\in\\oR^{n-k}\\atop\\|x\\|=1}p_{\\mu}(x)^2\\leq\\varepsilon\\|p\\|^2,$$ where $\\|p\\|$ is the Bombieri norm of $p$. We show that for each $d\\in\\oN$ and $\\varepsilon>0$ there exists $k_{d,\\varepsilon}$ such that for","authors_text":"Alexander Schrijver","cross_cats":["math.AC"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2012-11-15T10:46:47Z","title":"Low rank approximation of polynomials"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1211.3569","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:b4ba6ce9d34745332c0e010e7e8de0c32768a827bd35a75a5f03f472de7208cd","target":"record","created_at":"2026-05-18T03:40:39Z","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":"987ad78fa652a77982cf5a04d940c16349c462cf49272c1e31f36238000ceaf8","cross_cats_sorted":["math.AC"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2012-11-15T10:46:47Z","title_canon_sha256":"9b4f270199d29bef908aa93555384ca4602ce83d1ab3f95b85d912214026d0a6"},"schema_version":"1.0","source":{"id":"1211.3569","kind":"arxiv","version":1}},"canonical_sha256":"e3257d4e972443b7094e63db9339d03618dd53246ea103afd6e6ba94f2f4d96f","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"e3257d4e972443b7094e63db9339d03618dd53246ea103afd6e6ba94f2f4d96f","first_computed_at":"2026-05-18T03:40:39.397918Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:40:39.397918Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"MxFQ+b2vmmydKuKR+eQThsIdf+8iE5mDNaFL8MO/C+7+SvxqbAdJ07cyHdQtgNVgQPBrzm1SHxTBkgR99Nq3AA==","signature_status":"signed_v1","signed_at":"2026-05-18T03:40:39.398601Z","signed_message":"canonical_sha256_bytes"},"source_id":"1211.3569","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:b4ba6ce9d34745332c0e010e7e8de0c32768a827bd35a75a5f03f472de7208cd","sha256:5dd0ac8f3fb6e1e701897d963563f09ba5ccb4c9fcf050d19585e394f5c9e7c4"],"state_sha256":"822b6b7820a6005326bf0fb892fdf80d5ac1d2589ef5758a3f665c69fc3d8aa1"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"7QnmzpmywpJcDNcW3P1U2Hnet3T55/ceCSYYQb42DErYqDd05RfE0MswpQSO+4Wj3r/mJgT2sSpuiI8lyPqJCw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-27T14:44:56.468366Z","bundle_sha256":"c94e9e81ed5e851d9a941355bc2cab29cdb1f3513c58dedb2cd095f4a44039b7"}}