{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2016:4AV5UVIH7QUVROZXZUJL5UANOY","short_pith_number":"pith:4AV5UVIH","schema_version":"1.0","canonical_sha256":"e02bda5507fc2958bb37cd12bed00d76232d7c17839e5f271d0b84c9cdd42670","source":{"kind":"arxiv","id":"1611.09806","version":3},"attestation_state":"computed","paper":{"title":"Squarefree values of polynomial discriminants I","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Arul Shankar, Manjul Bhargava, Xiaoheng Wang","submitted_at":"2016-11-29T19:46:08Z","abstract_excerpt":"We determine the density of monic integer polynomials of given degree $n>1$ that have squarefree discriminant; in particular, we prove for the first time that the lower density of such polynomials is positive. Similarly, we prove that the density of monic integer polynomials $f(x)$, such that $f(x)$ is irreducible and $\\mathbb Z[x]/(f(x))$ is the ring of integers in its fraction field, is positive, and is in fact given by $\\zeta(2)^{-1}$.\n  It also follows from our methods that there are $\\gg X^{1/2+1/n}$ monogenic number fields of degree $n$ having associated Galois group $S_n$ and absolute d"},"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":"1611.09806","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2016-11-29T19:46:08Z","cross_cats_sorted":[],"title_canon_sha256":"18c61e4c29bd10702cd2fcee1f9b8ab00bec77c5453be13b52b4f20501f89028","abstract_canon_sha256":"f6cf08ec8aec8a6ebbd80306ea1d304355ea685781ca6ee97bcc8e3f84e0d3e9"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T03:44:48.186294Z","signature_b64":"SN5/+Z4jfQhKZ/2RJuJHKPnEFt1B4A4HL7z/7LdowUqSYzUSVSzlq0RRasXVbLiVUQJUlLOOErKcqUWsqSz3CA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"e02bda5507fc2958bb37cd12bed00d76232d7c17839e5f271d0b84c9cdd42670","last_reissued_at":"2026-07-05T03:44:48.185943Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T03:44:48.185943Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Squarefree values of polynomial discriminants I","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Arul Shankar, Manjul Bhargava, Xiaoheng Wang","submitted_at":"2016-11-29T19:46:08Z","abstract_excerpt":"We determine the density of monic integer polynomials of given degree $n>1$ that have squarefree discriminant; in particular, we prove for the first time that the lower density of such polynomials is positive. Similarly, we prove that the density of monic integer polynomials $f(x)$, such that $f(x)$ is irreducible and $\\mathbb Z[x]/(f(x))$ is the ring of integers in its fraction field, is positive, and is in fact given by $\\zeta(2)^{-1}$.\n  It also follows from our methods that there are $\\gg X^{1/2+1/n}$ monogenic number fields of degree $n$ having associated Galois group $S_n$ and absolute d"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1611.09806","kind":"arxiv","version":3},"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/1611.09806/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":"1611.09806","created_at":"2026-07-05T03:44:48.186006+00:00"},{"alias_kind":"arxiv_version","alias_value":"1611.09806v3","created_at":"2026-07-05T03:44:48.186006+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1611.09806","created_at":"2026-07-05T03:44:48.186006+00:00"},{"alias_kind":"pith_short_12","alias_value":"4AV5UVIH7QUV","created_at":"2026-07-05T03:44:48.186006+00:00"},{"alias_kind":"pith_short_16","alias_value":"4AV5UVIH7QUVROZX","created_at":"2026-07-05T03:44:48.186006+00:00"},{"alias_kind":"pith_short_8","alias_value":"4AV5UVIH","created_at":"2026-07-05T03:44:48.186006+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":1,"internal_anchor_count":1,"sample":[{"citing_arxiv_id":"1909.01338","citing_title":"A zero density estimate for Dedekind zeta functions","ref_index":1,"is_internal_anchor":true}]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/4AV5UVIH7QUVROZXZUJL5UANOY","json":"https://pith.science/pith/4AV5UVIH7QUVROZXZUJL5UANOY.json","graph_json":"https://pith.science/api/pith-number/4AV5UVIH7QUVROZXZUJL5UANOY/graph.json","events_json":"https://pith.science/api/pith-number/4AV5UVIH7QUVROZXZUJL5UANOY/events.json","paper":"https://pith.science/paper/4AV5UVIH"},"agent_actions":{"view_html":"https://pith.science/pith/4AV5UVIH7QUVROZXZUJL5UANOY","download_json":"https://pith.science/pith/4AV5UVIH7QUVROZXZUJL5UANOY.json","view_paper":"https://pith.science/paper/4AV5UVIH","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1611.09806&json=true","fetch_graph":"https://pith.science/api/pith-number/4AV5UVIH7QUVROZXZUJL5UANOY/graph.json","fetch_events":"https://pith.science/api/pith-number/4AV5UVIH7QUVROZXZUJL5UANOY/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/4AV5UVIH7QUVROZXZUJL5UANOY/action/timestamp_anchor","attest_storage":"https://pith.science/pith/4AV5UVIH7QUVROZXZUJL5UANOY/action/storage_attestation","attest_author":"https://pith.science/pith/4AV5UVIH7QUVROZXZUJL5UANOY/action/author_attestation","sign_citation":"https://pith.science/pith/4AV5UVIH7QUVROZXZUJL5UANOY/action/citation_signature","submit_replication":"https://pith.science/pith/4AV5UVIH7QUVROZXZUJL5UANOY/action/replication_record"}},"created_at":"2026-07-05T03:44:48.186006+00:00","updated_at":"2026-07-05T03:44:48.186006+00:00"}