{"paper":{"title":"A Real Nullstellensatz for Free Modules","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AG","authors_text":"Jaka Cimpric","submitted_at":"2013-02-10T20:23:27Z","abstract_excerpt":"Let $A$ be the algebra of all $n \\times n$ matrices with entries from $\\RR[x_1,\\ldots,x_d]$ and let $G_1,\\ldots,G_m,F \\in A$. We will show that $F(a)v=0$ for every $a \\in \\RR^d$ and $v \\in \\RR^n$ such that $G_i(a)v=0$ for all $i$ if and only if $F$ belongs to the smallest real left ideal of $A$ which contains $G_1,\\ldots,G_m$. Here a left ideal $J$ of $A$ is real if for every $H_1,\\ldots,H_k \\in A$ such that $H_1^T H_1+\\ldots+H_k^T H_k \\in J+J^T$ we have that $H_1,\\ldots,H_k \\in J$. We call this result the one-sided Real Nullstellensatz for matrix polynomials. We first prove by induction on $n"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1302.2358","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"}