{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2018:7AZPZIDZXHEC5VHBO6ZFDLQA4U","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":"7a86928ced4649c809e763baf634171a7dee9573f9f8634c2efd5d0fff1d6ff8","cross_cats_sorted":["cs.CC","cs.SC"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2018-08-30T21:50:17Z","title_canon_sha256":"805ac09ccc0f104b156effc19bf54a1357551a999cb41b0266b8ffcbd7c8d7ac"},"schema_version":"1.0","source":{"id":"1808.10531","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1808.10531","created_at":"2026-05-17T23:53:57Z"},{"alias_kind":"arxiv_version","alias_value":"1808.10531v2","created_at":"2026-05-17T23:53:57Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1808.10531","created_at":"2026-05-17T23:53:57Z"},{"alias_kind":"pith_short_12","alias_value":"7AZPZIDZXHEC","created_at":"2026-05-18T12:32:11Z"},{"alias_kind":"pith_short_16","alias_value":"7AZPZIDZXHEC5VHB","created_at":"2026-05-18T12:32:11Z"},{"alias_kind":"pith_short_8","alias_value":"7AZPZIDZ","created_at":"2026-05-18T12:32:11Z"}],"graph_snapshots":[{"event_id":"sha256:9a0c48097dba917ba147c2ae9f545555551662e504faa279d7b86ff873faba00","target":"graph","created_at":"2026-05-17T23:53:57Z","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":"Suppose $k,p\\!\\in\\!\\mathbb{N}$ with $p$ prime and $f\\!\\in\\!\\mathbb{Z}[x]$ is a univariate polynomial with degree $d$ and all coefficients having absolute value less than $p^k$. We give a Las Vegas randomized algorithm that computes the number of roots of $f$ in $\\mathbb{Z}/\\!\\left(p^k\\right)$ within time $d^3(k\\log p)^{2+o(1)}$. (We in fact prove a more intricate complexity bound that is slightly better.) The best previous general algorithm had (deterministic) complexity exponential in $k$. We also present some experimental data evincing the potential practicality of our algorithm.","authors_text":"J. Maurice Rojas, Leann Kopp, Natalie Randall, Yuyu Zhu","cross_cats":["cs.CC","cs.SC"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2018-08-30T21:50:17Z","title":"Randomized Polynomial-Time Root Counting in Prime Power Rings"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1808.10531","kind":"arxiv","version":2},"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":"sha256:7c2b45941fe05bb474bafbb7affc3a0354bab954717a48138c8a46e5eb1a4ca9","target":"record","created_at":"2026-05-17T23:53:57Z","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":"7a86928ced4649c809e763baf634171a7dee9573f9f8634c2efd5d0fff1d6ff8","cross_cats_sorted":["cs.CC","cs.SC"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2018-08-30T21:50:17Z","title_canon_sha256":"805ac09ccc0f104b156effc19bf54a1357551a999cb41b0266b8ffcbd7c8d7ac"},"schema_version":"1.0","source":{"id":"1808.10531","kind":"arxiv","version":2}},"canonical_sha256":"f832fca079b9c82ed4e177b251ae00e53e31447a2649a166b082a03b90550865","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"f832fca079b9c82ed4e177b251ae00e53e31447a2649a166b082a03b90550865","first_computed_at":"2026-05-17T23:53:57.533709Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-17T23:53:57.533709Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"gQwuWn5ditglW9CaUxQ+xpa3tb2CeY1awklZKyBN57raZ39ZC12nYiUDsIhm+SE1aL5vfFhYF2Pj/uE5yyAOCg==","signature_status":"signed_v1","signed_at":"2026-05-17T23:53:57.534399Z","signed_message":"canonical_sha256_bytes"},"source_id":"1808.10531","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:7c2b45941fe05bb474bafbb7affc3a0354bab954717a48138c8a46e5eb1a4ca9","sha256:9a0c48097dba917ba147c2ae9f545555551662e504faa279d7b86ff873faba00"],"state_sha256":"886889815e79081c589b810241aca42e77dea4b6446808cff213a43fa6a79ff4"}