{"paper":{"title":"Approximating Nonnegative Polynomials via Spectral Sparsification","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AG","math.MG"],"primary_cat":"math.OC","authors_text":"Alperen A. Erg\\\"ur","submitted_at":"2016-12-19T16:14:19Z","abstract_excerpt":"We study polyhedral approximations to the cone of nonnegative polynomials. We show that any constant ratio polyhedral approximation to the cone of nonnegative degree $2d$ forms in $n$ variables has to have exponentially many facets in terms of $n$. We also showthat for fixed $m \\geq 3$, all linear $m$ dimensional sections of the nonnegative cone that include $(x_1^2+x_2^2+\\ldots + x_n^2)^d$ has a costant ratio polyhedral approximation with $O(n^{m-2})$ many facets. Our approach is convex geometric, and parts of the argument rely on the recent solution of Kadison-Singer problem. We also discuss"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1612.06245","kind":"arxiv","version":5},"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"}