{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2015:VDVJCBR3LLBZTK36HCSGDHKHSL","short_pith_number":"pith:VDVJCBR3","canonical_record":{"source":{"id":"1509.05707","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2015-09-18T17:14:02Z","cross_cats_sorted":[],"title_canon_sha256":"7c534fbe9ea15d9f668ef202261f4367617b309da6d5c66e9896d2c803ad039e","abstract_canon_sha256":"e67850d66c8b4264eb344d298a25aeac29d600e4ce6aa1f5ea0abad8f5302c6a"},"schema_version":"1.0"},"canonical_sha256":"a8ea91063b5ac399ab7e38a4619d4792c8cb6ad0aec160b73a73c21f8cd1d4ff","source":{"kind":"arxiv","id":"1509.05707","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1509.05707","created_at":"2026-05-18T01:32:42Z"},{"alias_kind":"arxiv_version","alias_value":"1509.05707v1","created_at":"2026-05-18T01:32:42Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1509.05707","created_at":"2026-05-18T01:32:42Z"},{"alias_kind":"pith_short_12","alias_value":"VDVJCBR3LLBZ","created_at":"2026-05-18T12:29:44Z"},{"alias_kind":"pith_short_16","alias_value":"VDVJCBR3LLBZTK36","created_at":"2026-05-18T12:29:44Z"},{"alias_kind":"pith_short_8","alias_value":"VDVJCBR3","created_at":"2026-05-18T12:29:44Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2015:VDVJCBR3LLBZTK36HCSGDHKHSL","target":"record","payload":{"canonical_record":{"source":{"id":"1509.05707","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2015-09-18T17:14:02Z","cross_cats_sorted":[],"title_canon_sha256":"7c534fbe9ea15d9f668ef202261f4367617b309da6d5c66e9896d2c803ad039e","abstract_canon_sha256":"e67850d66c8b4264eb344d298a25aeac29d600e4ce6aa1f5ea0abad8f5302c6a"},"schema_version":"1.0"},"canonical_sha256":"a8ea91063b5ac399ab7e38a4619d4792c8cb6ad0aec160b73a73c21f8cd1d4ff","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:32:42.553261Z","signature_b64":"p8eWZoahTyca5eZYrh4IxT5sSaklAE1YchSx6Cur+B9fA7BowpDoK0pz67pL8KlZZMz2jPQa/gYEzI+j6EbwCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"a8ea91063b5ac399ab7e38a4619d4792c8cb6ad0aec160b73a73c21f8cd1d4ff","last_reissued_at":"2026-05-18T01:32:42.552735Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:32:42.552735Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1509.05707","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":"2026-05-18T01:32:42Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"xK299FXt6Ffr10fqa9vD+PivNfDyBSnF/BNUPjSoqzxgZfn33ku1dy/2oH6UcBLOlGAwRE0CMPxMqtfacKdmCA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-01T07:28:47.214357Z"},"content_sha256":"c16abf40efcbb80f9fd00fdc727878465adc5ff7e8a7002f93580f942028109f","schema_version":"1.0","event_id":"sha256:c16abf40efcbb80f9fd00fdc727878465adc5ff7e8a7002f93580f942028109f"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2015:VDVJCBR3LLBZTK36HCSGDHKHSL","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Symmetric multilinear forms and polarization of polynomials","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Ale\\v{s} Dr\\'apal, Petr Vojt\\v{e}chovsk\\'y","submitted_at":"2015-09-18T17:14:02Z","abstract_excerpt":"We study a generalization of the classical correspondence between homogeneous quadratic polynomials, quadratic forms, and symmetric/alternating bilinear forms to forms in $n$ variables. The main tool is combinatorial polarization, and the approach is applicable even when $n!$ is not invertible in the underlying field."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1509.05707","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":"2026-05-18T01:32:42Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"yXdY6DlaeYjihA8szYd0Wj4XxNaWRpBFA+AFmnoODgirfsBmhuj0ERR3kycOcoP6zb7HhUi+4hCNVF147F43Dg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-01T07:28:47.214702Z"},"content_sha256":"ae754c9b2265c42e1efbaf3bcf79fb9cdbca0182729a128e668834b0d0d02ccf","schema_version":"1.0","event_id":"sha256:ae754c9b2265c42e1efbaf3bcf79fb9cdbca0182729a128e668834b0d0d02ccf"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/VDVJCBR3LLBZTK36HCSGDHKHSL/bundle.json","state_url":"https://pith.science/pith/VDVJCBR3LLBZTK36HCSGDHKHSL/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/VDVJCBR3LLBZTK36HCSGDHKHSL/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":"2026-06-01T07:28:47Z","links":{"resolver":"https://pith.science/pith/VDVJCBR3LLBZTK36HCSGDHKHSL","bundle":"https://pith.science/pith/VDVJCBR3LLBZTK36HCSGDHKHSL/bundle.json","state":"https://pith.science/pith/VDVJCBR3LLBZTK36HCSGDHKHSL/state.json","well_known_bundle":"https://pith.science/.well-known/pith/VDVJCBR3LLBZTK36HCSGDHKHSL/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:VDVJCBR3LLBZTK36HCSGDHKHSL","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":"e67850d66c8b4264eb344d298a25aeac29d600e4ce6aa1f5ea0abad8f5302c6a","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2015-09-18T17:14:02Z","title_canon_sha256":"7c534fbe9ea15d9f668ef202261f4367617b309da6d5c66e9896d2c803ad039e"},"schema_version":"1.0","source":{"id":"1509.05707","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1509.05707","created_at":"2026-05-18T01:32:42Z"},{"alias_kind":"arxiv_version","alias_value":"1509.05707v1","created_at":"2026-05-18T01:32:42Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1509.05707","created_at":"2026-05-18T01:32:42Z"},{"alias_kind":"pith_short_12","alias_value":"VDVJCBR3LLBZ","created_at":"2026-05-18T12:29:44Z"},{"alias_kind":"pith_short_16","alias_value":"VDVJCBR3LLBZTK36","created_at":"2026-05-18T12:29:44Z"},{"alias_kind":"pith_short_8","alias_value":"VDVJCBR3","created_at":"2026-05-18T12:29:44Z"}],"graph_snapshots":[{"event_id":"sha256:ae754c9b2265c42e1efbaf3bcf79fb9cdbca0182729a128e668834b0d0d02ccf","target":"graph","created_at":"2026-05-18T01:32:42Z","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":"We study a generalization of the classical correspondence between homogeneous quadratic polynomials, quadratic forms, and symmetric/alternating bilinear forms to forms in $n$ variables. The main tool is combinatorial polarization, and the approach is applicable even when $n!$ is not invertible in the underlying field.","authors_text":"Ale\\v{s} Dr\\'apal, Petr Vojt\\v{e}chovsk\\'y","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2015-09-18T17:14:02Z","title":"Symmetric multilinear forms and polarization of polynomials"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1509.05707","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:c16abf40efcbb80f9fd00fdc727878465adc5ff7e8a7002f93580f942028109f","target":"record","created_at":"2026-05-18T01:32:42Z","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":"e67850d66c8b4264eb344d298a25aeac29d600e4ce6aa1f5ea0abad8f5302c6a","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2015-09-18T17:14:02Z","title_canon_sha256":"7c534fbe9ea15d9f668ef202261f4367617b309da6d5c66e9896d2c803ad039e"},"schema_version":"1.0","source":{"id":"1509.05707","kind":"arxiv","version":1}},"canonical_sha256":"a8ea91063b5ac399ab7e38a4619d4792c8cb6ad0aec160b73a73c21f8cd1d4ff","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"a8ea91063b5ac399ab7e38a4619d4792c8cb6ad0aec160b73a73c21f8cd1d4ff","first_computed_at":"2026-05-18T01:32:42.552735Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:32:42.552735Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"p8eWZoahTyca5eZYrh4IxT5sSaklAE1YchSx6Cur+B9fA7BowpDoK0pz67pL8KlZZMz2jPQa/gYEzI+j6EbwCg==","signature_status":"signed_v1","signed_at":"2026-05-18T01:32:42.553261Z","signed_message":"canonical_sha256_bytes"},"source_id":"1509.05707","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:c16abf40efcbb80f9fd00fdc727878465adc5ff7e8a7002f93580f942028109f","sha256:ae754c9b2265c42e1efbaf3bcf79fb9cdbca0182729a128e668834b0d0d02ccf"],"state_sha256":"e0f3ee2d0c910e955d415205bb5e134416356e4de26bb001f862c217ee4e0de3"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"pmzpJqj1UGufh+ej+BwXTaXqZzFRv+N4d70hbQgRcvaLh3Gs6ZmVYs1FOx/y+FVgWaW3SFKQ11oFH/podBa7AA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-01T07:28:47.216679Z","bundle_sha256":"c32d4fb421783addb8f2e5738f4b60a987abcc92c06723e3eb79ddc254cb9e80"}}