{"paper":{"title":"Do Sums of Squares Dream of Free Resolutions?","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AG","authors_text":"Grigoriy Blekherman, Mauricio Velasco, Rainer Sinn","submitted_at":"2016-07-13T00:19:34Z","abstract_excerpt":"We associate to a real projective variety $X$ two convex cones which are fundamental in real algebraic geometry: the cone $P_X$ of quadratic forms nonnegative on $X$, and the cone $\\Sigma_X$ of sums of squares of linear forms. The dual cone $\\Sigma_X^\\ast$ is a spectrahedron and we show that its convexity properties are closely related to homological properties of $X$. For instance, we show that all extreme rays of $\\Sigma_X^\\ast$ have rank one if and only if X has Castelnuovo-Mumford regularity two. More generally, if $\\Sigma_X^\\ast$ has an extreme ray of rank $p > 1$, then $X$ does not satis"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1607.03551","kind":"arxiv","version":2},"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"}