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Yet, since $\\mathbb{Q}\\_p$ is not an effective field, classical algorithm does not apply.We provide a definition for an approximate Gr{\\\"o}bner basis with respect to a monomial order $w.$ We design a strategy to compute such a basis, when precision is enough and under the assumption that the input sequence is regular and the ideals $\\langle f\\_1,\\dots,f\\_i \\rangle$ are weakly-$w$-ideals. The conjecture "},"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":"1403.5464","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.SC","submitted_at":"2014-03-20T12:37:01Z","cross_cats_sorted":["math.AC"],"title_canon_sha256":"dc48ae55366bc0bfcae33c7c800a0b207db3de8f50a255bcf2ea82d6600f83b2","abstract_canon_sha256":"5879841aff79c725b0b83c1ae41b3a19d41ce9c5f8a4b34b1e7989797e6433a0"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:32:08.504908Z","signature_b64":"6zCdiuhSbxiLfFEUarnOCMtn46CVrw2/UtwxyQU74ChWrQ3eo2Xi57e1lKZuhwy8qlOExQkpInuH9Uj3Dr9dCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"f15e1b0d4b9126cec99f7395fb719c26c1c5fb14c310b19c48881fc99cd6ff8b","last_reissued_at":"2026-05-18T01:32:08.504188Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:32:08.504188Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Matrix-F5 algorithms over finite-precision complete discrete valuation fields","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AC"],"primary_cat":"cs.SC","authors_text":"Tristan Vaccon (IRMAR)","submitted_at":"2014-03-20T12:37:01Z","abstract_excerpt":"Let $(f\\_1,\\dots, f\\_s) \\in \\mathbb{Q}\\_p [X\\_1,\\dots, X\\_n]^s$ be a sequence of homogeneous polynomials with $p$-adic coefficients. Such system may happen, for example, in arithmetic geometry. Yet, since $\\mathbb{Q}\\_p$ is not an effective field, classical algorithm does not apply.We provide a definition for an approximate Gr{\\\"o}bner basis with respect to a monomial order $w.$ We design a strategy to compute such a basis, when precision is enough and under the assumption that the input sequence is regular and the ideals $\\langle f\\_1,\\dots,f\\_i \\rangle$ are weakly-$w$-ideals. The conjecture "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1403.5464","kind":"arxiv","version":2},"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":"1403.5464","created_at":"2026-05-18T01:32:08.504289+00:00"},{"alias_kind":"arxiv_version","alias_value":"1403.5464v2","created_at":"2026-05-18T01:32:08.504289+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1403.5464","created_at":"2026-05-18T01:32:08.504289+00:00"},{"alias_kind":"pith_short_12","alias_value":"6FPBWDKLSETM","created_at":"2026-05-18T12:28:16.859392+00:00"},{"alias_kind":"pith_short_16","alias_value":"6FPBWDKLSETM5SM7","created_at":"2026-05-18T12:28:16.859392+00:00"},{"alias_kind":"pith_short_8","alias_value":"6FPBWDKL","created_at":"2026-05-18T12:28:16.859392+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/6FPBWDKLSETM5SM7OOK7W4M4E3","json":"https://pith.science/pith/6FPBWDKLSETM5SM7OOK7W4M4E3.json","graph_json":"https://pith.science/api/pith-number/6FPBWDKLSETM5SM7OOK7W4M4E3/graph.json","events_json":"https://pith.science/api/pith-number/6FPBWDKLSETM5SM7OOK7W4M4E3/events.json","paper":"https://pith.science/paper/6FPBWDKL"},"agent_actions":{"view_html":"https://pith.science/pith/6FPBWDKLSETM5SM7OOK7W4M4E3","download_json":"https://pith.science/pith/6FPBWDKLSETM5SM7OOK7W4M4E3.json","view_paper":"https://pith.science/paper/6FPBWDKL","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1403.5464&json=true","fetch_graph":"https://pith.science/api/pith-number/6FPBWDKLSETM5SM7OOK7W4M4E3/graph.json","fetch_events":"https://pith.science/api/pith-number/6FPBWDKLSETM5SM7OOK7W4M4E3/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/6FPBWDKLSETM5SM7OOK7W4M4E3/action/timestamp_anchor","attest_storage":"https://pith.science/pith/6FPBWDKLSETM5SM7OOK7W4M4E3/action/storage_attestation","attest_author":"https://pith.science/pith/6FPBWDKLSETM5SM7OOK7W4M4E3/action/author_attestation","sign_citation":"https://pith.science/pith/6FPBWDKLSETM5SM7OOK7W4M4E3/action/citation_signature","submit_replication":"https://pith.science/pith/6FPBWDKLSETM5SM7OOK7W4M4E3/action/replication_record"}},"created_at":"2026-05-18T01:32:08.504289+00:00","updated_at":"2026-05-18T01:32:08.504289+00:00"}