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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 "},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"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"},"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"},"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1407.6388","created_at":"2026-05-18T02:28:31.490247+00:00"},{"alias_kind":"arxiv_version","alias_value":"1407.6388v2","created_at":"2026-05-18T02:28:31.490247+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1407.6388","created_at":"2026-05-18T02:28:31.490247+00:00"},{"alias_kind":"pith_short_12","alias_value":"AJCMAOTCFEMN","created_at":"2026-05-18T12:28:19.803747+00:00"},{"alias_kind":"pith_short_16","alias_value":"AJCMAOTCFEMNCVLD","created_at":"2026-05-18T12:28:19.803747+00:00"},{"alias_kind":"pith_short_8","alias_value":"AJCMAOTC","created_at":"2026-05-18T12:28:19.803747+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/AJCMAOTCFEMNCVLD7OTGHP74LE","json":"https://pith.science/pith/AJCMAOTCFEMNCVLD7OTGHP74LE.json","graph_json":"https://pith.science/api/pith-number/AJCMAOTCFEMNCVLD7OTGHP74LE/graph.json","events_json":"https://pith.science/api/pith-number/AJCMAOTCFEMNCVLD7OTGHP74LE/events.json","paper":"https://pith.science/paper/AJCMAOTC"},"agent_actions":{"view_html":"https://pith.science/pith/AJCMAOTCFEMNCVLD7OTGHP74LE","download_json":"https://pith.science/pith/AJCMAOTCFEMNCVLD7OTGHP74LE.json","view_paper":"https://pith.science/paper/AJCMAOTC","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1407.6388&json=true","fetch_graph":"https://pith.science/api/pith-number/AJCMAOTCFEMNCVLD7OTGHP74LE/graph.json","fetch_events":"https://pith.science/api/pith-number/AJCMAOTCFEMNCVLD7OTGHP74LE/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/AJCMAOTCFEMNCVLD7OTGHP74LE/action/timestamp_anchor","attest_storage":"https://pith.science/pith/AJCMAOTCFEMNCVLD7OTGHP74LE/action/storage_attestation","attest_author":"https://pith.science/pith/AJCMAOTCFEMNCVLD7OTGHP74LE/action/author_attestation","sign_citation":"https://pith.science/pith/AJCMAOTCFEMNCVLD7OTGHP74LE/action/citation_signature","submit_replication":"https://pith.science/pith/AJCMAOTCFEMNCVLD7OTGHP74LE/action/replication_record"}},"created_at":"2026-05-18T02:28:31.490247+00:00","updated_at":"2026-05-18T02:28:31.490247+00:00"}