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For $1 \\leq i \\leq \\ell$, denote by $\\mathcal{Q}_i = \\{Q_1, \\ldots, Q_i \\}$, $V_i$ the real variety defined by $\\mathcal{Q}_i$, and $k_i$ an upper bound on the real dimension of $V_i$ (by convention $V_0 = \\mathrm{R}^k$ and $k_0 = k$). Suppose also that \\[ 2 \\leq d_1 \\leq d_2 \\leq \\frac{1}{k + 1} d_3 \\leq \\frac{1}{(k + 1)^2} d_4 \\leq \\cdots \\leq \\frac{1}{(k + 1)^{\\ell - 3}} d_{\\ell - 1} \\leq \\frac{1}{(k + 1)^{\\ell - 2}} d"},"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":"1303.1577","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2013-03-07T00:14:26Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"f7ad8d22c6ca4d925351b0eb044eafd4a54664c2e267dc5c45793bd3393327d1","abstract_canon_sha256":"165ae1801cd00fc310425768217b07e33c6d6e911e098f08f1ede86bfe92bce9"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:32:19.197554Z","signature_b64":"kMxnR6i1SpOqJ50sryeZyIIIvQSBa8WWhDo2MErqbNS+YQOtQCJnShila+FSLHJjz4MWIlEYJP1YtSuCVsZ2BA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"f2916a7793a17229725bc64503420751566d3084775f017b807eff04990f9402","last_reissued_at":"2026-05-18T01:32:19.196786Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:32:19.196786Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On a real analogue of Bezout inequality and the number of connected components of sign conditions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.AG","authors_text":"Sal Barone, Saugata Basu","submitted_at":"2013-03-07T00:14:26Z","abstract_excerpt":"Let $\\mathrm{R}$ be a real closed field and $Q_1, \\ldots, Q_{\\ell} \\in \\mathrm{R}[X_1, \\ldots,X_k]$ such that for each $i, 1 \\leq i \\leq \\ell$, $\\mathrm{deg} (Q_i) \\leq d_i$. For $1 \\leq i \\leq \\ell$, denote by $\\mathcal{Q}_i = \\{Q_1, \\ldots, Q_i \\}$, $V_i$ the real variety defined by $\\mathcal{Q}_i$, and $k_i$ an upper bound on the real dimension of $V_i$ (by convention $V_0 = \\mathrm{R}^k$ and $k_0 = k$). Suppose also that \\[ 2 \\leq d_1 \\leq d_2 \\leq \\frac{1}{k + 1} d_3 \\leq \\frac{1}{(k + 1)^2} d_4 \\leq \\cdots \\leq \\frac{1}{(k + 1)^{\\ell - 3}} d_{\\ell - 1} \\leq \\frac{1}{(k + 1)^{\\ell - 2}} d"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1303.1577","kind":"arxiv","version":4},"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":"1303.1577","created_at":"2026-05-18T01:32:19.196928+00:00"},{"alias_kind":"arxiv_version","alias_value":"1303.1577v4","created_at":"2026-05-18T01:32:19.196928+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1303.1577","created_at":"2026-05-18T01:32:19.196928+00:00"},{"alias_kind":"pith_short_12","alias_value":"6KIWU54TUFZC","created_at":"2026-05-18T12:27:36.564083+00:00"},{"alias_kind":"pith_short_16","alias_value":"6KIWU54TUFZCS4S3","created_at":"2026-05-18T12:27:36.564083+00:00"},{"alias_kind":"pith_short_8","alias_value":"6KIWU54T","created_at":"2026-05-18T12:27:36.564083+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/6KIWU54TUFZCS4S3YZCQGQQHKF","json":"https://pith.science/pith/6KIWU54TUFZCS4S3YZCQGQQHKF.json","graph_json":"https://pith.science/api/pith-number/6KIWU54TUFZCS4S3YZCQGQQHKF/graph.json","events_json":"https://pith.science/api/pith-number/6KIWU54TUFZCS4S3YZCQGQQHKF/events.json","paper":"https://pith.science/paper/6KIWU54T"},"agent_actions":{"view_html":"https://pith.science/pith/6KIWU54TUFZCS4S3YZCQGQQHKF","download_json":"https://pith.science/pith/6KIWU54TUFZCS4S3YZCQGQQHKF.json","view_paper":"https://pith.science/paper/6KIWU54T","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1303.1577&json=true","fetch_graph":"https://pith.science/api/pith-number/6KIWU54TUFZCS4S3YZCQGQQHKF/graph.json","fetch_events":"https://pith.science/api/pith-number/6KIWU54TUFZCS4S3YZCQGQQHKF/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/6KIWU54TUFZCS4S3YZCQGQQHKF/action/timestamp_anchor","attest_storage":"https://pith.science/pith/6KIWU54TUFZCS4S3YZCQGQQHKF/action/storage_attestation","attest_author":"https://pith.science/pith/6KIWU54TUFZCS4S3YZCQGQQHKF/action/author_attestation","sign_citation":"https://pith.science/pith/6KIWU54TUFZCS4S3YZCQGQQHKF/action/citation_signature","submit_replication":"https://pith.science/pith/6KIWU54TUFZCS4S3YZCQGQQHKF/action/replication_record"}},"created_at":"2026-05-18T01:32:19.196928+00:00","updated_at":"2026-05-18T01:32:19.196928+00:00"}