{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2007:MLKE3USSNOUJZDRN4UPQE6VL7H","short_pith_number":"pith:MLKE3USS","canonical_record":{"source":{"id":"math/0702773","kind":"arxiv","version":1},"metadata":{"license":"","primary_cat":"math.CO","submitted_at":"2007-02-26T13:12:07Z","cross_cats_sorted":[],"title_canon_sha256":"ec082bec9ffd429fa23c33b28846d0288c1999e1090ca7afa9aee1590ccac212","abstract_canon_sha256":"6e91dceeb4be732481c1dd233d1ec8a4ee7ce6a45bd54c206106cbea16ba5af7"},"schema_version":"1.0"},"canonical_sha256":"62d44dd2526ba89c8e2de51f027aabf9eb2c35ffb8045713ff463203fc0a2895","source":{"kind":"arxiv","id":"math/0702773","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"math/0702773","created_at":"2026-05-18T04:28:20Z"},{"alias_kind":"arxiv_version","alias_value":"math/0702773v1","created_at":"2026-05-18T04:28:20Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/0702773","created_at":"2026-05-18T04:28:20Z"},{"alias_kind":"pith_short_12","alias_value":"MLKE3USSNOUJ","created_at":"2026-05-18T12:25:55Z"},{"alias_kind":"pith_short_16","alias_value":"MLKE3USSNOUJZDRN","created_at":"2026-05-18T12:25:55Z"},{"alias_kind":"pith_short_8","alias_value":"MLKE3USS","created_at":"2026-05-18T12:25:55Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2007:MLKE3USSNOUJZDRN4UPQE6VL7H","target":"record","payload":{"canonical_record":{"source":{"id":"math/0702773","kind":"arxiv","version":1},"metadata":{"license":"","primary_cat":"math.CO","submitted_at":"2007-02-26T13:12:07Z","cross_cats_sorted":[],"title_canon_sha256":"ec082bec9ffd429fa23c33b28846d0288c1999e1090ca7afa9aee1590ccac212","abstract_canon_sha256":"6e91dceeb4be732481c1dd233d1ec8a4ee7ce6a45bd54c206106cbea16ba5af7"},"schema_version":"1.0"},"canonical_sha256":"62d44dd2526ba89c8e2de51f027aabf9eb2c35ffb8045713ff463203fc0a2895","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:28:20.587390Z","signature_b64":"FItNc11nOketbGtm6fbCui+auyKpjcuwoEWg53Cm0ZQJATiZhhOoENkJB2ASRgJlYP1UamU7JHpi6miEIc70Cg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"62d44dd2526ba89c8e2de51f027aabf9eb2c35ffb8045713ff463203fc0a2895","last_reissued_at":"2026-05-18T04:28:20.586950Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:28:20.586950Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"math/0702773","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":"2026-05-18T04:28:20Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"vgDNcIGFID1lE/P/SXYv1sHl3KYJW92OtY9n+ru217jt0ANz+CLGFbTs7NJFY2bQDHJEzipm2JnSIYjM3DT7AQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-29T14:32:50.601594Z"},"content_sha256":"93f79a2e821db18c0b3bcd19a140fe345e328d60173bab9bd575d5870007e1db","schema_version":"1.0","event_id":"sha256:93f79a2e821db18c0b3bcd19a140fe345e328d60173bab9bd575d5870007e1db"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2007:MLKE3USSNOUJZDRN4UPQE6VL7H","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Polynomials that Sign Represent Parity and Descartes' Rule of Signs","license":"","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Nayantara Bhatnagar, Parikshit Gopalan, Richard J. Lipton, Saugata Basu","submitted_at":"2007-02-26T13:12:07Z","abstract_excerpt":"A real polynomial $P(X_1,..., X_n)$ sign represents $f: A^n \\to \\{0,1\\}$ if for every $(a_1, ..., a_n) \\in A^n$, the sign of $P(a_1,...,a_n)$ equals $(-1)^{f(a_1,...,a_n)}$. Such sign representations are well-studied in computer science and have applications to computational complexity and computational learning theory. In this work, we present a systematic study of tradeoffs between degree and sparsity of sign representations through the lens of the parity function. We attempt to prove bounds that hold for any choice of set $A$. We show that sign representing parity over $\\{0,...,m-1\\}^n$ wit"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0702773","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":"2026-05-18T04:28:20Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"LgrR6YP0iiLJSapGcBsejlwM7o5SLIlu2wlx9UuAeI0bDnykG7aPEPHRHvs6rin7oZGqvsJsh96pyUrLYH+yAA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-29T14:32:50.602271Z"},"content_sha256":"dfed795ff0ef4dec90652288e1d5262b77e1b7c3d0f757410962d94c14402a95","schema_version":"1.0","event_id":"sha256:dfed795ff0ef4dec90652288e1d5262b77e1b7c3d0f757410962d94c14402a95"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/MLKE3USSNOUJZDRN4UPQE6VL7H/bundle.json","state_url":"https://pith.science/pith/MLKE3USSNOUJZDRN4UPQE6VL7H/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/MLKE3USSNOUJZDRN4UPQE6VL7H/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":"2026-05-29T14:32:50Z","links":{"resolver":"https://pith.science/pith/MLKE3USSNOUJZDRN4UPQE6VL7H","bundle":"https://pith.science/pith/MLKE3USSNOUJZDRN4UPQE6VL7H/bundle.json","state":"https://pith.science/pith/MLKE3USSNOUJZDRN4UPQE6VL7H/state.json","well_known_bundle":"https://pith.science/.well-known/pith/MLKE3USSNOUJZDRN4UPQE6VL7H/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2007:MLKE3USSNOUJZDRN4UPQE6VL7H","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":"6e91dceeb4be732481c1dd233d1ec8a4ee7ce6a45bd54c206106cbea16ba5af7","cross_cats_sorted":[],"license":"","primary_cat":"math.CO","submitted_at":"2007-02-26T13:12:07Z","title_canon_sha256":"ec082bec9ffd429fa23c33b28846d0288c1999e1090ca7afa9aee1590ccac212"},"schema_version":"1.0","source":{"id":"math/0702773","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"math/0702773","created_at":"2026-05-18T04:28:20Z"},{"alias_kind":"arxiv_version","alias_value":"math/0702773v1","created_at":"2026-05-18T04:28:20Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/0702773","created_at":"2026-05-18T04:28:20Z"},{"alias_kind":"pith_short_12","alias_value":"MLKE3USSNOUJ","created_at":"2026-05-18T12:25:55Z"},{"alias_kind":"pith_short_16","alias_value":"MLKE3USSNOUJZDRN","created_at":"2026-05-18T12:25:55Z"},{"alias_kind":"pith_short_8","alias_value":"MLKE3USS","created_at":"2026-05-18T12:25:55Z"}],"graph_snapshots":[{"event_id":"sha256:dfed795ff0ef4dec90652288e1d5262b77e1b7c3d0f757410962d94c14402a95","target":"graph","created_at":"2026-05-18T04:28:20Z","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 real polynomial $P(X_1,..., X_n)$ sign represents $f: A^n \\to \\{0,1\\}$ if for every $(a_1, ..., a_n) \\in A^n$, the sign of $P(a_1,...,a_n)$ equals $(-1)^{f(a_1,...,a_n)}$. Such sign representations are well-studied in computer science and have applications to computational complexity and computational learning theory. In this work, we present a systematic study of tradeoffs between degree and sparsity of sign representations through the lens of the parity function. We attempt to prove bounds that hold for any choice of set $A$. We show that sign representing parity over $\\{0,...,m-1\\}^n$ wit","authors_text":"Nayantara Bhatnagar, Parikshit Gopalan, Richard J. Lipton, Saugata Basu","cross_cats":[],"headline":"","license":"","primary_cat":"math.CO","submitted_at":"2007-02-26T13:12:07Z","title":"Polynomials that Sign Represent Parity and Descartes' Rule of Signs"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0702773","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":"sha256:93f79a2e821db18c0b3bcd19a140fe345e328d60173bab9bd575d5870007e1db","target":"record","created_at":"2026-05-18T04:28:20Z","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":"6e91dceeb4be732481c1dd233d1ec8a4ee7ce6a45bd54c206106cbea16ba5af7","cross_cats_sorted":[],"license":"","primary_cat":"math.CO","submitted_at":"2007-02-26T13:12:07Z","title_canon_sha256":"ec082bec9ffd429fa23c33b28846d0288c1999e1090ca7afa9aee1590ccac212"},"schema_version":"1.0","source":{"id":"math/0702773","kind":"arxiv","version":1}},"canonical_sha256":"62d44dd2526ba89c8e2de51f027aabf9eb2c35ffb8045713ff463203fc0a2895","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"62d44dd2526ba89c8e2de51f027aabf9eb2c35ffb8045713ff463203fc0a2895","first_computed_at":"2026-05-18T04:28:20.586950Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:28:20.586950Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"FItNc11nOketbGtm6fbCui+auyKpjcuwoEWg53Cm0ZQJATiZhhOoENkJB2ASRgJlYP1UamU7JHpi6miEIc70Cg==","signature_status":"signed_v1","signed_at":"2026-05-18T04:28:20.587390Z","signed_message":"canonical_sha256_bytes"},"source_id":"math/0702773","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:93f79a2e821db18c0b3bcd19a140fe345e328d60173bab9bd575d5870007e1db","sha256:dfed795ff0ef4dec90652288e1d5262b77e1b7c3d0f757410962d94c14402a95"],"state_sha256":"f4d69f33a8ce0dc36eaf06d00855404532151c3ee870801382907ac5bddde0ae"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"iFzzQNnQF+j1GOIqlolyh4jiCVHNU3KzASMChlfXmB5cLO8P+h3ni3XcxW/D5g01cK+YyAjl/hauE0cuVIpjDA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-29T14:32:50.605793Z","bundle_sha256":"40b0e80090587bc1061a76648a8b75dc10fd4adbeba56b940501f1595f96a427"}}