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1}.$ Under the assumption that the map $F \\colon {\\Bbb R}^n \\rightarrow {\\Bbb R}^p$ is convenient and non-degenerate at infinity, we show that there exists a constant $c > 0$ such that the following so-called {\\em H\\\"older-type global error bound result} holds $$c d(x,S) \\le [f(x)]_+^{\\frac{2}{\\mathcal{H}(2d, n, p)}} + [f(x)]"},"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":"1411.0859","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OC","submitted_at":"2014-11-04T10:57:47Z","cross_cats_sorted":[],"title_canon_sha256":"eea1fa99a4e17a9f3f832c1f5b1a78187c6c1f7a1f36120f160240d1f0bb71f9","abstract_canon_sha256":"f53cf275056b69c05f8e3c122c3000f7fa1d2b96f210ee39c976e12b413ceea5"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:38:38.072866Z","signature_b64":"RtgpBAOx1x5DuiV+tthS3Y0YaUigExUQ+sOLZ3cpe0ZMlK8q7IBG3KkFRfoHcWm9WKNtsnmaTYoGz06v9KnOAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"1c5e520d178e38139e65f875ee922f3050f1c606c157a0d6e370610defb821aa","last_reissued_at":"2026-05-18T02:38:38.072202Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:38:38.072202Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"H\\\"older-Type Global Error Bounds for Non-degenerate Polynomial Systems","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.OC","authors_text":"Ha Huy Vui, Pham Tien Son, Si Tiep Dinh","submitted_at":"2014-11-04T10:57:47Z","abstract_excerpt":"Let $F := (f_1, \\ldots, f_p) \\colon {\\Bbb R}^n \\to {\\Bbb R}^p$ be a polynomial map, and suppose that $S := \\{x \\in {\\Bbb R}^n \\ : \\ f_i(x) \\le 0, i = 1, \\ldots, p\\} \\ne \\emptyset.$ Let $d := \\max_{i = 1, \\ldots, p} \\deg f_i$ and $\\mathcal{H}(d, n, p) := d(6d - 3)^{n + p - 1}.$ Under the assumption that the map $F \\colon {\\Bbb R}^n \\rightarrow {\\Bbb R}^p$ is convenient and non-degenerate at infinity, we show that there exists a constant $c > 0$ such that the following so-called {\\em H\\\"older-type global error bound result} holds $$c d(x,S) \\le [f(x)]_+^{\\frac{2}{\\mathcal{H}(2d, n, p)}} + [f(x)]"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1411.0859","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":"1411.0859","created_at":"2026-05-18T02:38:38.072311+00:00"},{"alias_kind":"arxiv_version","alias_value":"1411.0859v1","created_at":"2026-05-18T02:38:38.072311+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1411.0859","created_at":"2026-05-18T02:38:38.072311+00:00"},{"alias_kind":"pith_short_12","alias_value":"DRPFEDIXRY4B","created_at":"2026-05-18T12:28:25.294606+00:00"},{"alias_kind":"pith_short_16","alias_value":"DRPFEDIXRY4BHHTF","created_at":"2026-05-18T12:28:25.294606+00:00"},{"alias_kind":"pith_short_8","alias_value":"DRPFEDIX","created_at":"2026-05-18T12:28:25.294606+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/DRPFEDIXRY4BHHTF7B265ERPGB","json":"https://pith.science/pith/DRPFEDIXRY4BHHTF7B265ERPGB.json","graph_json":"https://pith.science/api/pith-number/DRPFEDIXRY4BHHTF7B265ERPGB/graph.json","events_json":"https://pith.science/api/pith-number/DRPFEDIXRY4BHHTF7B265ERPGB/events.json","paper":"https://pith.science/paper/DRPFEDIX"},"agent_actions":{"view_html":"https://pith.science/pith/DRPFEDIXRY4BHHTF7B265ERPGB","download_json":"https://pith.science/pith/DRPFEDIXRY4BHHTF7B265ERPGB.json","view_paper":"https://pith.science/paper/DRPFEDIX","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1411.0859&json=true","fetch_graph":"https://pith.science/api/pith-number/DRPFEDIXRY4BHHTF7B265ERPGB/graph.json","fetch_events":"https://pith.science/api/pith-number/DRPFEDIXRY4BHHTF7B265ERPGB/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/DRPFEDIXRY4BHHTF7B265ERPGB/action/timestamp_anchor","attest_storage":"https://pith.science/pith/DRPFEDIXRY4BHHTF7B265ERPGB/action/storage_attestation","attest_author":"https://pith.science/pith/DRPFEDIXRY4BHHTF7B265ERPGB/action/author_attestation","sign_citation":"https://pith.science/pith/DRPFEDIXRY4BHHTF7B265ERPGB/action/citation_signature","submit_replication":"https://pith.science/pith/DRPFEDIXRY4BHHTF7B265ERPGB/action/replication_record"}},"created_at":"2026-05-18T02:38:38.072311+00:00","updated_at":"2026-05-18T02:38:38.072311+00:00"}