{"paper":{"title":"Generic matrix polynomials with fixed rank and fixed degree","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NA","authors_text":"Andrii Dmytryshyn, Froil\\'an M. Dopico","submitted_at":"2016-12-13T10:19:27Z","abstract_excerpt":"The set ${\\cal P}^{m\\times n}_{r,d}$ of $m \\times n$ complex matrix polynomials of grade $d$ and (normal) rank at most $r$ in a complex $(d+1)mn$ dimensional space is studied. For $r = 1, \\dots , \\min \\{m, n\\}-1$, we show that ${\\cal P}^{m\\times n}_{r,d}$ is the union of the closures of the $rd+1$ sets of matrix polynomials with rank $r$, degree exactly $d$, and explicitly described complete eigenstructures. In addition, for the full-rank rectangular polynomials, i.e. $r= \\min \\{m, n\\}$ and $m \\neq n$, we show that ${\\cal P}^{m\\times n}_{r,d}$ coincides with the closure of a single set of the "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1612.04085","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"}