{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2015:HT7QCCJWSLO6UM5KGVVNEY5HJC","short_pith_number":"pith:HT7QCCJW","canonical_record":{"source":{"id":"1501.05767","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2015-01-23T11:05:33Z","cross_cats_sorted":[],"title_canon_sha256":"ee287c7c79c069fa1279d1aaf92a180ac710f20e62ee794fbdeb686e2bf6f8ae","abstract_canon_sha256":"b0f208560ac645a5bc578c85a897d17b2d8ccfc6a68cf05967fe08522cd55010"},"schema_version":"1.0"},"canonical_sha256":"3cff01093692ddea33aa356ad263a7489d35097ea79f234db1c1d1b7ca3928d1","source":{"kind":"arxiv","id":"1501.05767","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1501.05767","created_at":"2026-05-18T02:28:49Z"},{"alias_kind":"arxiv_version","alias_value":"1501.05767v1","created_at":"2026-05-18T02:28:49Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1501.05767","created_at":"2026-05-18T02:28:49Z"},{"alias_kind":"pith_short_12","alias_value":"HT7QCCJWSLO6","created_at":"2026-05-18T12:29:25Z"},{"alias_kind":"pith_short_16","alias_value":"HT7QCCJWSLO6UM5K","created_at":"2026-05-18T12:29:25Z"},{"alias_kind":"pith_short_8","alias_value":"HT7QCCJW","created_at":"2026-05-18T12:29:25Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2015:HT7QCCJWSLO6UM5KGVVNEY5HJC","target":"record","payload":{"canonical_record":{"source":{"id":"1501.05767","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2015-01-23T11:05:33Z","cross_cats_sorted":[],"title_canon_sha256":"ee287c7c79c069fa1279d1aaf92a180ac710f20e62ee794fbdeb686e2bf6f8ae","abstract_canon_sha256":"b0f208560ac645a5bc578c85a897d17b2d8ccfc6a68cf05967fe08522cd55010"},"schema_version":"1.0"},"canonical_sha256":"3cff01093692ddea33aa356ad263a7489d35097ea79f234db1c1d1b7ca3928d1","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:28:49.045888Z","signature_b64":"RwbiG204H86cYPJx/PZwgYs/aq7QPHAV1VoDO7wEtmLjjigOD1Fd2nZdcGrVVqx62DSBzNAu+o7ug6Y6hRuhBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"3cff01093692ddea33aa356ad263a7489d35097ea79f234db1c1d1b7ca3928d1","last_reissued_at":"2026-05-18T02:28:49.045525Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:28:49.045525Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1501.05767","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T02:28:49Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"+DE3i3Vj4Z+hqhr1hahTIN9szv5Gg4epG6tr5mvbr1o3Y0aQxwdFBKdnFZazxaLE+flCELK+xTHEzBLvkfuNDQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-25T08:10:53.081175Z"},"content_sha256":"70876e395390ed396b4a4c667f087ff3276acfb0c9a8f2fdb16e152ad0e03a78","schema_version":"1.0","event_id":"sha256:70876e395390ed396b4a4c667f087ff3276acfb0c9a8f2fdb16e152ad0e03a78"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2015:HT7QCCJWSLO6UM5KGVVNEY5HJC","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Integral polynomials with small discriminants and resultants","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Friedrich G\\\"otze, Vasili Bernik, Victor Beresnevich","submitted_at":"2015-01-23T11:05:33Z","abstract_excerpt":"Let $n\\in\\mathbb{N}$ be fixed, $Q>1$ be a real parameter and $\\mathcal{P}_n(Q)$ denote the set of polynomials over $\\mathbb{Z}$ of degree $n$ and height at most $Q$. In this paper we investigate the following counting problems regarding polynomials with small discriminant $D(P)$ and pairs of polynomials with small resultant $R(P_1,P_2)$:\n  (i) given $0\\le v\\le n-1$ and a sufficiently large $Q$, estimate the number of polynomials $P\\in\\mathcal{P}_n(Q)$ such that $$0<|D(P)|\\le Q^{2n-2-2v};$$ (ii) given $0\\le w\\le n$ and a sufficiently large $Q$, estimate the number of pairs of polynomials $P_1,P"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1501.05767","kind":"arxiv","version":1},"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":"2026-05-18T02:28:49Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"DGPAny9fjngFHDSMWCLDkp2tsP0OqsFdD3HOtc6rU/BILdImQ/Ts3reFeeR7krjo6c+bV72nsd/MkOsIelIFAQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-25T08:10:53.081537Z"},"content_sha256":"51b35a716a82acc345bcedefa4560845ac1e4774bd5483d1834135347e08f0d1","schema_version":"1.0","event_id":"sha256:51b35a716a82acc345bcedefa4560845ac1e4774bd5483d1834135347e08f0d1"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/HT7QCCJWSLO6UM5KGVVNEY5HJC/bundle.json","state_url":"https://pith.science/pith/HT7QCCJWSLO6UM5KGVVNEY5HJC/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/HT7QCCJWSLO6UM5KGVVNEY5HJC/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":"2026-06-25T08:10:53Z","links":{"resolver":"https://pith.science/pith/HT7QCCJWSLO6UM5KGVVNEY5HJC","bundle":"https://pith.science/pith/HT7QCCJWSLO6UM5KGVVNEY5HJC/bundle.json","state":"https://pith.science/pith/HT7QCCJWSLO6UM5KGVVNEY5HJC/state.json","well_known_bundle":"https://pith.science/.well-known/pith/HT7QCCJWSLO6UM5KGVVNEY5HJC/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:HT7QCCJWSLO6UM5KGVVNEY5HJC","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":"b0f208560ac645a5bc578c85a897d17b2d8ccfc6a68cf05967fe08522cd55010","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2015-01-23T11:05:33Z","title_canon_sha256":"ee287c7c79c069fa1279d1aaf92a180ac710f20e62ee794fbdeb686e2bf6f8ae"},"schema_version":"1.0","source":{"id":"1501.05767","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1501.05767","created_at":"2026-05-18T02:28:49Z"},{"alias_kind":"arxiv_version","alias_value":"1501.05767v1","created_at":"2026-05-18T02:28:49Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1501.05767","created_at":"2026-05-18T02:28:49Z"},{"alias_kind":"pith_short_12","alias_value":"HT7QCCJWSLO6","created_at":"2026-05-18T12:29:25Z"},{"alias_kind":"pith_short_16","alias_value":"HT7QCCJWSLO6UM5K","created_at":"2026-05-18T12:29:25Z"},{"alias_kind":"pith_short_8","alias_value":"HT7QCCJW","created_at":"2026-05-18T12:29:25Z"}],"graph_snapshots":[{"event_id":"sha256:51b35a716a82acc345bcedefa4560845ac1e4774bd5483d1834135347e08f0d1","target":"graph","created_at":"2026-05-18T02:28:49Z","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 $n\\in\\mathbb{N}$ be fixed, $Q>1$ be a real parameter and $\\mathcal{P}_n(Q)$ denote the set of polynomials over $\\mathbb{Z}$ of degree $n$ and height at most $Q$. In this paper we investigate the following counting problems regarding polynomials with small discriminant $D(P)$ and pairs of polynomials with small resultant $R(P_1,P_2)$:\n  (i) given $0\\le v\\le n-1$ and a sufficiently large $Q$, estimate the number of polynomials $P\\in\\mathcal{P}_n(Q)$ such that $$0<|D(P)|\\le Q^{2n-2-2v};$$ (ii) given $0\\le w\\le n$ and a sufficiently large $Q$, estimate the number of pairs of polynomials $P_1,P","authors_text":"Friedrich G\\\"otze, Vasili Bernik, Victor Beresnevich","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2015-01-23T11:05:33Z","title":"Integral polynomials with small discriminants and resultants"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1501.05767","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":"sha256:70876e395390ed396b4a4c667f087ff3276acfb0c9a8f2fdb16e152ad0e03a78","target":"record","created_at":"2026-05-18T02:28:49Z","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":"b0f208560ac645a5bc578c85a897d17b2d8ccfc6a68cf05967fe08522cd55010","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2015-01-23T11:05:33Z","title_canon_sha256":"ee287c7c79c069fa1279d1aaf92a180ac710f20e62ee794fbdeb686e2bf6f8ae"},"schema_version":"1.0","source":{"id":"1501.05767","kind":"arxiv","version":1}},"canonical_sha256":"3cff01093692ddea33aa356ad263a7489d35097ea79f234db1c1d1b7ca3928d1","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"3cff01093692ddea33aa356ad263a7489d35097ea79f234db1c1d1b7ca3928d1","first_computed_at":"2026-05-18T02:28:49.045525Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:28:49.045525Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"RwbiG204H86cYPJx/PZwgYs/aq7QPHAV1VoDO7wEtmLjjigOD1Fd2nZdcGrVVqx62DSBzNAu+o7ug6Y6hRuhBw==","signature_status":"signed_v1","signed_at":"2026-05-18T02:28:49.045888Z","signed_message":"canonical_sha256_bytes"},"source_id":"1501.05767","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:70876e395390ed396b4a4c667f087ff3276acfb0c9a8f2fdb16e152ad0e03a78","sha256:51b35a716a82acc345bcedefa4560845ac1e4774bd5483d1834135347e08f0d1"],"state_sha256":"65e2c0dd78af9094dfa283eaadaa65099c82f5a6be8cdbb14a53dc3e19a1948d"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"WQOeXn5r7BFErHGTnyxyCurluzVtAa+AnPjp6muA8P4baN/aD9rB6kR/51nbBqupS+CR9F8pd6KnA8I9wGLQAg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-25T08:10:53.083583Z","bundle_sha256":"4c6597252351333e2d67f14aef059dc85ea79ffeacef6adcfe2d686f2eb77630"}}