{"paper":{"title":"Log-concavity and lower bounds for arithmetic circuits","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.DM","math.AC"],"primary_cat":"cs.CC","authors_text":"Ignacio Garc\\'ia-Marco (LIP), Pascal Koiran (LIP), S\\'ebastien Tavenas","submitted_at":"2015-03-26T12:29:43Z","abstract_excerpt":"One question that we investigate in this paper is, how can we build log-concave polynomials using sparse polynomials as building blocks? More precisely, let $f = \\sum\\_{i = 0}^d a\\_i X^i \\in \\mathbb{R}^+[X]$ be a polynomial satisfying the log-concavity condition $a\\_i^2 \\textgreater{} \\tau a\\_{i-1}a\\_{i+1}$ for every $i \\in \\{1,\\ldots,d-1\\},$ where $\\tau \\textgreater{} 0$. Whenever $f$ can be written under the form $f = \\sum\\_{i = 1}^k \\prod\\_{j = 1}^m f\\_{i,j}$ where the polynomials $f\\_{i,j}$ have at most $t$ monomials, it is clear that $d \\leq k t^m$. Assuming that the $f\\_{i,j}$ have only "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1503.07705","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"}