{"paper":{"title":"Free loci of matrix pencils and domains of noncommutative rational functions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.RT"],"primary_cat":"math.RA","authors_text":"Igor Klep, Jurij Vol\\v{c}i\\v{c}","submitted_at":"2015-12-08T21:00:11Z","abstract_excerpt":"Consider a monic linear pencil $L(x) = I - A_1x_1 - \\cdots - A_gx_g$ whose coefficients $A_j$ are $d \\times d$ matrices. It is naturally evaluated at $g$-tuples of matrices $X$ using the Kronecker tensor product, which gives rise to its free locus $Z(L) = \\{ X: \\det L(X) = 0 \\}$. In this article it is shown that the algebras $A$ and $A'$ generated by the coefficients of two linear pencils $L$ and $L'$, respectively, with equal free loci are isomorphic up to radical. Furthermore, $Z(L) \\subseteq Z(L')$ if and only if the natural map sending the coefficients of $L'$ to the coefficients of $L$ in"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1512.02648","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":""},"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"}