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Suppose $f$ is $s$-sparse, has an individual degree of at most $d$, and a total degree of $D = \\tdeg(f)$. We prove a sparsity bound on the base polynomial $g$: \\[ \\|g\\|_0 \\le s^{D(2d+2)/e + 1}. \\] Based on this bound, we develop a deterministic algorithm that computes the base $g$. %\nIn contrast to the general deterministic factorization algorithm of Bhargava, Saraf, and Volkov"},"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":"2607.02364","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DS","submitted_at":"2026-07-02T16:06:29Z","cross_cats_sorted":["math.AC"],"title_canon_sha256":"47703ff8f8ea275d83d4ceb1aa3d791658708f500219f1372d64ec5917edd864","abstract_canon_sha256":"3988a217b5f67b16d18579758b69680e4c6655174be9713ec5ba743dcbcfed71"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-03T01:17:56.778163Z","signature_b64":"mak8XrSbaiUVvoxwbgL9DT1L8O1vyNYmBFctgNPSSPVpBDgZ7qvzaNJ4cx0HUxjKIuvuzUHCEgp1okE6g6rHDQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"5ac4000050f0ad7058face1f9324eef1591e43b30a1a62073f6f702b1c373892","last_reissued_at":"2026-07-03T01:17:56.777723Z","signature_status":"signed_v1","first_computed_at":"2026-07-03T01:17:56.777723Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Deterministic Polynomial-time Exact-root Computation for Sparse Polynomials with Bounded Total Degree","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AC"],"primary_cat":"cs.DS","authors_text":"Qiao-Long Huang, Ruichen Qiu, Xiao-Shan Gao, Yichuan Cao","submitted_at":"2026-07-02T16:06:29Z","abstract_excerpt":"We study the problem of deterministically computing the exact root of a sparse polynomial in the multivariate setting. 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We prove a sparsity bound on the base polynomial $g$: \\[ \\|g\\|_0 \\le s^{D(2d+2)/e + 1}. \\] Based on this bound, we develop a deterministic algorithm that computes the base $g$. %\nIn contrast to the general deterministic factorization algorithm of Bhargava, Saraf, and Volkov"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.02364","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2607.02364/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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":"2607.02364","created_at":"2026-07-03T01:17:56.777774+00:00"},{"alias_kind":"arxiv_version","alias_value":"2607.02364v1","created_at":"2026-07-03T01:17:56.777774+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2607.02364","created_at":"2026-07-03T01:17:56.777774+00:00"},{"alias_kind":"pith_short_12","alias_value":"LLCAAACQ6CWX","created_at":"2026-07-03T01:17:56.777774+00:00"},{"alias_kind":"pith_short_16","alias_value":"LLCAAACQ6CWXAWH2","created_at":"2026-07-03T01:17:56.777774+00:00"},{"alias_kind":"pith_short_8","alias_value":"LLCAAACQ","created_at":"2026-07-03T01:17:56.777774+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/LLCAAACQ6CWXAWH2ZYPZGJHO6F","json":"https://pith.science/pith/LLCAAACQ6CWXAWH2ZYPZGJHO6F.json","graph_json":"https://pith.science/api/pith-number/LLCAAACQ6CWXAWH2ZYPZGJHO6F/graph.json","events_json":"https://pith.science/api/pith-number/LLCAAACQ6CWXAWH2ZYPZGJHO6F/events.json","paper":"https://pith.science/paper/LLCAAACQ"},"agent_actions":{"view_html":"https://pith.science/pith/LLCAAACQ6CWXAWH2ZYPZGJHO6F","download_json":"https://pith.science/pith/LLCAAACQ6CWXAWH2ZYPZGJHO6F.json","view_paper":"https://pith.science/paper/LLCAAACQ","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2607.02364&json=true","fetch_graph":"https://pith.science/api/pith-number/LLCAAACQ6CWXAWH2ZYPZGJHO6F/graph.json","fetch_events":"https://pith.science/api/pith-number/LLCAAACQ6CWXAWH2ZYPZGJHO6F/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/LLCAAACQ6CWXAWH2ZYPZGJHO6F/action/timestamp_anchor","attest_storage":"https://pith.science/pith/LLCAAACQ6CWXAWH2ZYPZGJHO6F/action/storage_attestation","attest_author":"https://pith.science/pith/LLCAAACQ6CWXAWH2ZYPZGJHO6F/action/author_attestation","sign_citation":"https://pith.science/pith/LLCAAACQ6CWXAWH2ZYPZGJHO6F/action/citation_signature","submit_replication":"https://pith.science/pith/LLCAAACQ6CWXAWH2ZYPZGJHO6F/action/replication_record"}},"created_at":"2026-07-03T01:17:56.777774+00:00","updated_at":"2026-07-03T01:17:56.777774+00:00"}