{"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"}