{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2017:BWDFWEFCOOMJPJC2HRGVZGGOXA","short_pith_number":"pith:BWDFWEFC","schema_version":"1.0","canonical_sha256":"0d865b10a2739897a45a3c4d5c98ceb821d90d379cf1f9757ca5af3dfc34240a","source":{"kind":"arxiv","id":"1711.01355","version":1},"attestation_state":"computed","paper":{"title":"Counting Roots of Polynomials Over Prime Power Rings","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.CC","cs.SC"],"primary_cat":"math.NT","authors_text":"Daqing Wan, J. Maurice Rojas, Qi Cheng, Shuhong Gao","submitted_at":"2017-11-03T23:02:22Z","abstract_excerpt":"Suppose $p$ is a prime, $t$ is a positive integer, and $f\\!\\in\\!\\mathbb{Z}[x]$ is a univariate polynomial of degree $d$ with coefficients of absolute value $<\\!p^t$. We show that for any fixed $t$, we can compute the number of roots in $\\mathbb{Z}/(p^t)$ of $f$ in deterministic time $(d+\\log p)^{O(1)}$. This fixed parameter tractability appears to be new for $t\\!\\geq\\!3$. A consequence for arithmetic geometry is that we can efficiently compute Igusa zeta functions $Z$, for univariate polynomials, assuming the degree of $Z$ is fixed."},"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":"1711.01355","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2017-11-03T23:02:22Z","cross_cats_sorted":["cs.CC","cs.SC"],"title_canon_sha256":"ba3bc267444ce5a46e9b54e3b0f9bc868778ce22690d0993ee778f6c698d5417","abstract_canon_sha256":"dc07d791124c2516d05dd54fb2cfaedf4953ee53192c920b5133436e494a5cba"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:54:11.732115Z","signature_b64":"bd/7hZJyy3vOHGkKk7Hdf9K07v+FvxTzl3rCSG3tpg/T4n/yRs7Dvjbbyk7xoOOjJG7+hCKm9vRniNHw5/0PCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"0d865b10a2739897a45a3c4d5c98ceb821d90d379cf1f9757ca5af3dfc34240a","last_reissued_at":"2026-05-17T23:54:11.731386Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:54:11.731386Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Counting Roots of Polynomials Over Prime Power Rings","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.CC","cs.SC"],"primary_cat":"math.NT","authors_text":"Daqing Wan, J. Maurice Rojas, Qi Cheng, Shuhong Gao","submitted_at":"2017-11-03T23:02:22Z","abstract_excerpt":"Suppose $p$ is a prime, $t$ is a positive integer, and $f\\!\\in\\!\\mathbb{Z}[x]$ is a univariate polynomial of degree $d$ with coefficients of absolute value $<\\!p^t$. We show that for any fixed $t$, we can compute the number of roots in $\\mathbb{Z}/(p^t)$ of $f$ in deterministic time $(d+\\log p)^{O(1)}$. This fixed parameter tractability appears to be new for $t\\!\\geq\\!3$. A consequence for arithmetic geometry is that we can efficiently compute Igusa zeta functions $Z$, for univariate polynomials, assuming the degree of $Z$ is fixed."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1711.01355","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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1711.01355","created_at":"2026-05-17T23:54:11.731517+00:00"},{"alias_kind":"arxiv_version","alias_value":"1711.01355v1","created_at":"2026-05-17T23:54:11.731517+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1711.01355","created_at":"2026-05-17T23:54:11.731517+00:00"},{"alias_kind":"pith_short_12","alias_value":"BWDFWEFCOOMJ","created_at":"2026-05-18T12:31:08.081275+00:00"},{"alias_kind":"pith_short_16","alias_value":"BWDFWEFCOOMJPJC2","created_at":"2026-05-18T12:31:08.081275+00:00"},{"alias_kind":"pith_short_8","alias_value":"BWDFWEFC","created_at":"2026-05-18T12:31:08.081275+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/BWDFWEFCOOMJPJC2HRGVZGGOXA","json":"https://pith.science/pith/BWDFWEFCOOMJPJC2HRGVZGGOXA.json","graph_json":"https://pith.science/api/pith-number/BWDFWEFCOOMJPJC2HRGVZGGOXA/graph.json","events_json":"https://pith.science/api/pith-number/BWDFWEFCOOMJPJC2HRGVZGGOXA/events.json","paper":"https://pith.science/paper/BWDFWEFC"},"agent_actions":{"view_html":"https://pith.science/pith/BWDFWEFCOOMJPJC2HRGVZGGOXA","download_json":"https://pith.science/pith/BWDFWEFCOOMJPJC2HRGVZGGOXA.json","view_paper":"https://pith.science/paper/BWDFWEFC","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1711.01355&json=true","fetch_graph":"https://pith.science/api/pith-number/BWDFWEFCOOMJPJC2HRGVZGGOXA/graph.json","fetch_events":"https://pith.science/api/pith-number/BWDFWEFCOOMJPJC2HRGVZGGOXA/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/BWDFWEFCOOMJPJC2HRGVZGGOXA/action/timestamp_anchor","attest_storage":"https://pith.science/pith/BWDFWEFCOOMJPJC2HRGVZGGOXA/action/storage_attestation","attest_author":"https://pith.science/pith/BWDFWEFCOOMJPJC2HRGVZGGOXA/action/author_attestation","sign_citation":"https://pith.science/pith/BWDFWEFCOOMJPJC2HRGVZGGOXA/action/citation_signature","submit_replication":"https://pith.science/pith/BWDFWEFCOOMJPJC2HRGVZGGOXA/action/replication_record"}},"created_at":"2026-05-17T23:54:11.731517+00:00","updated_at":"2026-05-17T23:54:11.731517+00:00"}