{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2014:AJCMAOTCFEMNCVLD7OTGHP74LE","short_pith_number":"pith:AJCMAOTC","canonical_record":{"source":{"id":"1407.6388","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2014-07-23T20:56:09Z","cross_cats_sorted":["math.PR"],"title_canon_sha256":"e9b3385ceaed8eb57e7416fd0d905e4abd4293acc3bac02e4e0b0a13e8252bae","abstract_canon_sha256":"3133dbaa19360eb385415944289018fadade972cb2de7355c79c2f701105c40c"},"schema_version":"1.0"},"canonical_sha256":"0244c03a622918d15563fba663bffc5925dcfc39f24d0c3881c564ec859582e2","source":{"kind":"arxiv","id":"1407.6388","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1407.6388","created_at":"2026-05-18T02:28:31Z"},{"alias_kind":"arxiv_version","alias_value":"1407.6388v2","created_at":"2026-05-18T02:28:31Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1407.6388","created_at":"2026-05-18T02:28:31Z"},{"alias_kind":"pith_short_12","alias_value":"AJCMAOTCFEMN","created_at":"2026-05-18T12:28:19Z"},{"alias_kind":"pith_short_16","alias_value":"AJCMAOTCFEMNCVLD","created_at":"2026-05-18T12:28:19Z"},{"alias_kind":"pith_short_8","alias_value":"AJCMAOTC","created_at":"2026-05-18T12:28:19Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2014:AJCMAOTCFEMNCVLD7OTGHP74LE","target":"record","payload":{"canonical_record":{"source":{"id":"1407.6388","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2014-07-23T20:56:09Z","cross_cats_sorted":["math.PR"],"title_canon_sha256":"e9b3385ceaed8eb57e7416fd0d905e4abd4293acc3bac02e4e0b0a13e8252bae","abstract_canon_sha256":"3133dbaa19360eb385415944289018fadade972cb2de7355c79c2f701105c40c"},"schema_version":"1.0"},"canonical_sha256":"0244c03a622918d15563fba663bffc5925dcfc39f24d0c3881c564ec859582e2","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:28:31.490679Z","signature_b64":"h4EveI/tf1J560AIbAC4nuAaQJ47CvkYKmQcgOwY1oTAlGe1j7dleBF6ljgFBwRvvQs+stzhc/74LkxcwYqOAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"0244c03a622918d15563fba663bffc5925dcfc39f24d0c3881c564ec859582e2","last_reissued_at":"2026-05-18T02:28:31.490173Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:28:31.490173Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1407.6388","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T02:28:31Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"EKkeTR0NTNQ/YDdi3urSI5NiHWy3Bdg5PZ4ETB/kiS7VcE+YaFUWk2Tz4HxqPDU4iNTaJnL/zBYD7N6DjiNMAA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-26T16:04:58.026278Z"},"content_sha256":"9732a57aec43cd3d3de395ce13cb9a0a5c2bba51a69c397171e9c99c44a9ae53","schema_version":"1.0","event_id":"sha256:9732a57aec43cd3d3de395ce13cb9a0a5c2bba51a69c397171e9c99c44a9ae53"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2014:AJCMAOTCFEMNCVLD7OTGHP74LE","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Discriminant and root separation of integral polynomials","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.PR"],"primary_cat":"math.NT","authors_text":"Dmitry Zaporozhets, Friedrich G\\\"otze","submitted_at":"2014-07-23T20:56:09Z","abstract_excerpt":"Consider a random polynomial $$ G_Q(x)=\\xi_{Q,n}x^n+\\xi_{Q,n-1}x^{n-1}+...+\\xi_{Q,0} $$ with independent coefficients uniformly distributed on $2Q+1$ integer points $\\{-Q, ..., Q\\}$. Denote by $D(G_Q)$ the discriminant of $G_Q$. We show that there exists a constant $C_n$, depending on $n$ only such that for all $Q\\ge 2$ the distribution of $D(G_Q)$ can be approximated as follows $$ \\sup_{-\\infty\\leq a\\leq b\\leq\\infty}|\\mathbb{P}(a\\leq \\frac{D(G_Q)}{Q^{2n-2}}\\leq b)-\\int_a^b\\varphi_n(x)\\, dx|\\leq\\frac{C_n}{\\log Q}, $$ where $\\varphi_n$ denotes the distribution function of the discriminant of a "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1407.6388","kind":"arxiv","version":2},"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":"2026-05-18T02:28:31Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"pxtJVlnESyd+4TU3+m4895+Nk5Bs3LRh70kOuUhTKeEi2SsjJ078rz6ay9ZBIxiVMppy4ENgEmZ64C6ZAEWiAw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-26T16:04:58.027017Z"},"content_sha256":"19cbf7937e314fcc710e181378ca019c3c3abaa9192d7df1b36560203ec776c3","schema_version":"1.0","event_id":"sha256:19cbf7937e314fcc710e181378ca019c3c3abaa9192d7df1b36560203ec776c3"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/AJCMAOTCFEMNCVLD7OTGHP74LE/bundle.json","state_url":"https://pith.science/pith/AJCMAOTCFEMNCVLD7OTGHP74LE/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/AJCMAOTCFEMNCVLD7OTGHP74LE/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":"2026-05-26T16:04:58Z","links":{"resolver":"https://pith.science/pith/AJCMAOTCFEMNCVLD7OTGHP74LE","bundle":"https://pith.science/pith/AJCMAOTCFEMNCVLD7OTGHP74LE/bundle.json","state":"https://pith.science/pith/AJCMAOTCFEMNCVLD7OTGHP74LE/state.json","well_known_bundle":"https://pith.science/.well-known/pith/AJCMAOTCFEMNCVLD7OTGHP74LE/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2014:AJCMAOTCFEMNCVLD7OTGHP74LE","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":"3133dbaa19360eb385415944289018fadade972cb2de7355c79c2f701105c40c","cross_cats_sorted":["math.PR"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2014-07-23T20:56:09Z","title_canon_sha256":"e9b3385ceaed8eb57e7416fd0d905e4abd4293acc3bac02e4e0b0a13e8252bae"},"schema_version":"1.0","source":{"id":"1407.6388","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1407.6388","created_at":"2026-05-18T02:28:31Z"},{"alias_kind":"arxiv_version","alias_value":"1407.6388v2","created_at":"2026-05-18T02:28:31Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1407.6388","created_at":"2026-05-18T02:28:31Z"},{"alias_kind":"pith_short_12","alias_value":"AJCMAOTCFEMN","created_at":"2026-05-18T12:28:19Z"},{"alias_kind":"pith_short_16","alias_value":"AJCMAOTCFEMNCVLD","created_at":"2026-05-18T12:28:19Z"},{"alias_kind":"pith_short_8","alias_value":"AJCMAOTC","created_at":"2026-05-18T12:28:19Z"}],"graph_snapshots":[{"event_id":"sha256:19cbf7937e314fcc710e181378ca019c3c3abaa9192d7df1b36560203ec776c3","target":"graph","created_at":"2026-05-18T02:28:31Z","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":"Consider a random polynomial $$ G_Q(x)=\\xi_{Q,n}x^n+\\xi_{Q,n-1}x^{n-1}+...+\\xi_{Q,0} $$ with independent coefficients uniformly distributed on $2Q+1$ integer points $\\{-Q, ..., Q\\}$. Denote by $D(G_Q)$ the discriminant of $G_Q$. We show that there exists a constant $C_n$, depending on $n$ only such that for all $Q\\ge 2$ the distribution of $D(G_Q)$ can be approximated as follows $$ \\sup_{-\\infty\\leq a\\leq b\\leq\\infty}|\\mathbb{P}(a\\leq \\frac{D(G_Q)}{Q^{2n-2}}\\leq b)-\\int_a^b\\varphi_n(x)\\, dx|\\leq\\frac{C_n}{\\log Q}, $$ where $\\varphi_n$ denotes the distribution function of the discriminant of a ","authors_text":"Dmitry Zaporozhets, Friedrich G\\\"otze","cross_cats":["math.PR"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2014-07-23T20:56:09Z","title":"Discriminant and root separation of integral polynomials"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1407.6388","kind":"arxiv","version":2},"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":"sha256:9732a57aec43cd3d3de395ce13cb9a0a5c2bba51a69c397171e9c99c44a9ae53","target":"record","created_at":"2026-05-18T02:28:31Z","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":"3133dbaa19360eb385415944289018fadade972cb2de7355c79c2f701105c40c","cross_cats_sorted":["math.PR"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2014-07-23T20:56:09Z","title_canon_sha256":"e9b3385ceaed8eb57e7416fd0d905e4abd4293acc3bac02e4e0b0a13e8252bae"},"schema_version":"1.0","source":{"id":"1407.6388","kind":"arxiv","version":2}},"canonical_sha256":"0244c03a622918d15563fba663bffc5925dcfc39f24d0c3881c564ec859582e2","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"0244c03a622918d15563fba663bffc5925dcfc39f24d0c3881c564ec859582e2","first_computed_at":"2026-05-18T02:28:31.490173Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:28:31.490173Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"h4EveI/tf1J560AIbAC4nuAaQJ47CvkYKmQcgOwY1oTAlGe1j7dleBF6ljgFBwRvvQs+stzhc/74LkxcwYqOAQ==","signature_status":"signed_v1","signed_at":"2026-05-18T02:28:31.490679Z","signed_message":"canonical_sha256_bytes"},"source_id":"1407.6388","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:9732a57aec43cd3d3de395ce13cb9a0a5c2bba51a69c397171e9c99c44a9ae53","sha256:19cbf7937e314fcc710e181378ca019c3c3abaa9192d7df1b36560203ec776c3"],"state_sha256":"bd069f6ff7b99e57a38a4a6fa3ea67f8bdc14b3d8a7f65fb2354f33e306118c9"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Lp9HD0uS396tNiKs9tVyRLi/x/rBhYVkLponq2xfO5Ed8MGAPiflbYlaSI1wlZkLJLFgCDjzx5syaTjj1G/wAQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-26T16:04:58.029869Z","bundle_sha256":"c792228889e0a909e0820e60f1df79cd8fd6cac95040347eac5d6b609cc46903"}}