{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2019:KMXA23TLGP5H2HJ53PXJMYVMRW","short_pith_number":"pith:KMXA23TL","schema_version":"1.0","canonical_sha256":"532e0d6e6b33fa7d1d3ddbee9662ac8d8c6ee0d16e2590f736e66337e2081b0c","source":{"kind":"arxiv","id":"1909.00135","version":2},"attestation_state":"computed","paper":{"title":"Discriminants of Fields Generated by Polynomials of Given Height","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Alina Ostafe, Igor E. Shparlinski, Rainer Dietmann","submitted_at":"2019-08-31T05:44:09Z","abstract_excerpt":"We obtain upper bounds for the number of monic irreducible polynomials over $\\mathbb Z$ of a fixed degree $n$ and a growing height $H$ for which the field generated by one of its roots has a given discriminant. We approach it via counting square-free parts of polynomial discriminants via two complementing approaches. In turn, this leads to a lower bound on the number of distinct discriminants of fields generated by roots of polynomials of degree $n$ and height at most $H$. We also give an upper bound for the number of trinomials of bounded height with given square-free part of the discriminant"},"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":"1909.00135","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2019-08-31T05:44:09Z","cross_cats_sorted":[],"title_canon_sha256":"ebba8162ef0ccaddd862ca1cc862968a94f262b90bca4b46f614f0d8739ea28d","abstract_canon_sha256":"39de710308d41d7f958298431238e9d3c0bafbdfb2efe754424f5a2bacbdcf3d"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T03:32:48.120363Z","signature_b64":"+lVxGSIUWfltF6rTsx6J9OdgcQr07P1QVbusjP06ELLq1IaprAU9/rtCWg3iJxn1qW+qsqEdUagS7Ld6tYt+Dg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"532e0d6e6b33fa7d1d3ddbee9662ac8d8c6ee0d16e2590f736e66337e2081b0c","last_reissued_at":"2026-07-05T03:32:48.119893Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T03:32:48.119893Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Discriminants of Fields Generated by Polynomials of Given Height","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Alina Ostafe, Igor E. Shparlinski, Rainer Dietmann","submitted_at":"2019-08-31T05:44:09Z","abstract_excerpt":"We obtain upper bounds for the number of monic irreducible polynomials over $\\mathbb Z$ of a fixed degree $n$ and a growing height $H$ for which the field generated by one of its roots has a given discriminant. We approach it via counting square-free parts of polynomial discriminants via two complementing approaches. In turn, this leads to a lower bound on the number of distinct discriminants of fields generated by roots of polynomials of degree $n$ and height at most $H$. We also give an upper bound for the number of trinomials of bounded height with given square-free part of the discriminant"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1909.00135","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/1909.00135/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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":"1909.00135","created_at":"2026-07-05T03:32:48.119951+00:00"},{"alias_kind":"arxiv_version","alias_value":"1909.00135v2","created_at":"2026-07-05T03:32:48.119951+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1909.00135","created_at":"2026-07-05T03:32:48.119951+00:00"},{"alias_kind":"pith_short_12","alias_value":"KMXA23TLGP5H","created_at":"2026-07-05T03:32:48.119951+00:00"},{"alias_kind":"pith_short_16","alias_value":"KMXA23TLGP5H2HJ5","created_at":"2026-07-05T03:32:48.119951+00:00"},{"alias_kind":"pith_short_8","alias_value":"KMXA23TL","created_at":"2026-07-05T03:32:48.119951+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":1,"internal_anchor_count":0,"sample":[{"citing_arxiv_id":"2304.11991","citing_title":"Enumerative Galois theory for number fields","ref_index":25,"is_internal_anchor":false}]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/KMXA23TLGP5H2HJ53PXJMYVMRW","json":"https://pith.science/pith/KMXA23TLGP5H2HJ53PXJMYVMRW.json","graph_json":"https://pith.science/api/pith-number/KMXA23TLGP5H2HJ53PXJMYVMRW/graph.json","events_json":"https://pith.science/api/pith-number/KMXA23TLGP5H2HJ53PXJMYVMRW/events.json","paper":"https://pith.science/paper/KMXA23TL"},"agent_actions":{"view_html":"https://pith.science/pith/KMXA23TLGP5H2HJ53PXJMYVMRW","download_json":"https://pith.science/pith/KMXA23TLGP5H2HJ53PXJMYVMRW.json","view_paper":"https://pith.science/paper/KMXA23TL","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1909.00135&json=true","fetch_graph":"https://pith.science/api/pith-number/KMXA23TLGP5H2HJ53PXJMYVMRW/graph.json","fetch_events":"https://pith.science/api/pith-number/KMXA23TLGP5H2HJ53PXJMYVMRW/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/KMXA23TLGP5H2HJ53PXJMYVMRW/action/timestamp_anchor","attest_storage":"https://pith.science/pith/KMXA23TLGP5H2HJ53PXJMYVMRW/action/storage_attestation","attest_author":"https://pith.science/pith/KMXA23TLGP5H2HJ53PXJMYVMRW/action/author_attestation","sign_citation":"https://pith.science/pith/KMXA23TLGP5H2HJ53PXJMYVMRW/action/citation_signature","submit_replication":"https://pith.science/pith/KMXA23TLGP5H2HJ53PXJMYVMRW/action/replication_record"}},"created_at":"2026-07-05T03:32:48.119951+00:00","updated_at":"2026-07-05T03:32:48.119951+00:00"}