{"paper":{"title":"A Real Nullstellensatz for Matrices of Non-Commutative Polynomials","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.OA","authors_text":"Christopher S. Nelson","submitted_at":"2013-05-03T18:39:14Z","abstract_excerpt":"This article extends the classical Real Nullstellensatz to matrices of polynomials in a free $\\ast$-algebra $\\RR\\axs$ with $x=(x_1, \\ldots, x_n)$. This result is a generalization of a result of Cimpri\\vc, Helton, McCullough, and the author.\n  In the free left $\\RR\\axs$-module $\\RR^{1 \\times \\ell}\\axs$ we introduce notions of the (noncommutative) zero set of a left $\\RR\\axs$-submodule and of a real left $\\RR\\axs$-submodule. We prove that every element from $\\RR^{1 \\times \\ell}\\axs$ whose zero set contains the intersection of zero sets of elements from a finite subset $S \\subset \\RR^{1 \\times \\e"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1305.0799","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"}