{"paper":{"title":"A Positivstellensatz for Conditional SAGE Signomials","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.LG","math.AG"],"primary_cat":"math.OC","authors_text":"Allen Houze Wang, Pascal Poupart, Priyank Jaini, Yaoliang Yu","submitted_at":"2020-03-08T06:20:10Z","abstract_excerpt":"Recently, the conditional SAGE certificate has been proposed as a sufficient condition for signomial positivity over a convex set. In this article, we show that the conditional SAGE certificate is $\\textit{complete}$. That is, for any signomial $f(\\mathbf{x}) = \\sum_{j=1}^{\\ell}c_j \\exp(\\mathbf{A}_j\\mathbf{x})$ defined by rational exponents that is positive over a compact convex set $\\mathcal{X}$, there is $p \\in \\mathbb{Z}_+$ and a specific positive definite function $w(\\mathbf{x})$ such that $w(\\mathbf{x})^p f(\\mathbf{x})$ may be verified by the conditional SAGE certificate. The completeness"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2003.03731","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2003.03731/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}