{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2011:HY47KIQVBCT3UHOMPOA6TFGDLE","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":"f9baf2b9f708437a1c4eb58166c7439d69d933d10f9de375ac214f67ae49d948","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2011-06-07T16:07:39Z","title_canon_sha256":"c378368fbc27b6ad4c8e0ee5a6f54d79693a6d6b06056effdab493358d74989f"},"schema_version":"1.0","source":{"id":"1106.1385","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1106.1385","created_at":"2026-05-18T04:20:23Z"},{"alias_kind":"arxiv_version","alias_value":"1106.1385v1","created_at":"2026-05-18T04:20:23Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1106.1385","created_at":"2026-05-18T04:20:23Z"},{"alias_kind":"pith_short_12","alias_value":"HY47KIQVBCT3","created_at":"2026-05-18T12:26:30Z"},{"alias_kind":"pith_short_16","alias_value":"HY47KIQVBCT3UHOM","created_at":"2026-05-18T12:26:30Z"},{"alias_kind":"pith_short_8","alias_value":"HY47KIQV","created_at":"2026-05-18T12:26:30Z"}],"graph_snapshots":[{"event_id":"sha256:0d5aa77836bb44813dbd8e7bd37bc7615000df36eaf19276fed21a5b3fc2990e","target":"graph","created_at":"2026-05-18T04:20:23Z","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":"Let $k$ be a number field, $f(x)\\in k[x]$ a polynomial over $k$ with $f(0)\\neq 0$, and $\\O_{k,S}^*$ the group of $S$-units of $k$, where $S$ is an appropriate finite set of places of $k$. In this note, we prove that outside of some natural exceptional set $T\\subset \\O_{k,S}^*$, the prime ideals of $\\O_k$ dividing $f(u)$, $u\\in \\O_{k,S}^*\\setminus T$, mostly have degree one over $\\Q$; that is, the corresponding residue fields have degree one over the prime field. We also formulate a conjectural analogue of this result for rational points on an elliptic curve over a number field, and deduce our ","authors_text":"Aaron Levin, David McKinnon","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2011-06-07T16:07:39Z","title":"Ideals of degree one contribute most of the height"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1106.1385","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:c2cb2b94a26115cfced21b6708a9200b4dd9c4c143032eab78311cbbc5ae3481","target":"record","created_at":"2026-05-18T04:20:23Z","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":"f9baf2b9f708437a1c4eb58166c7439d69d933d10f9de375ac214f67ae49d948","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2011-06-07T16:07:39Z","title_canon_sha256":"c378368fbc27b6ad4c8e0ee5a6f54d79693a6d6b06056effdab493358d74989f"},"schema_version":"1.0","source":{"id":"1106.1385","kind":"arxiv","version":1}},"canonical_sha256":"3e39f5221508a7ba1dcc7b81e994c3593903018cd5a1efea0808b248c3222e44","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"3e39f5221508a7ba1dcc7b81e994c3593903018cd5a1efea0808b248c3222e44","first_computed_at":"2026-05-18T04:20:23.798988Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:20:23.798988Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"t2z2JUnmqubE88jgYWih2kvoNrGAfkifVJXovlZfv1awkPLHWNmRk0Lab2TzCOR1Os2buE61Rrf30/0iuR/4Dg==","signature_status":"signed_v1","signed_at":"2026-05-18T04:20:23.799775Z","signed_message":"canonical_sha256_bytes"},"source_id":"1106.1385","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:c2cb2b94a26115cfced21b6708a9200b4dd9c4c143032eab78311cbbc5ae3481","sha256:0d5aa77836bb44813dbd8e7bd37bc7615000df36eaf19276fed21a5b3fc2990e"],"state_sha256":"57893521a6d98e11e28b7a3bf5e777d2e32e02081e9cc413a679f3095fad6cc8"}