{"paper":{"title":"Topology of tensor ranks","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AT","math.NA"],"primary_cat":"math.AG","authors_text":"Ke Ye, Lek-Heng Lim, Pierre Comon, Yang Qi","submitted_at":"2018-04-22T02:45:34Z","abstract_excerpt":"We study path-connectedness and homotopy groups of sets of tensors defined by tensor rank, border rank, multilinear rank, as well as their symmetric counterparts for symmetric tensors. We show that over $\\mathbb{C}$, the set of rank-$r$ tensors and the set of symmetric rank-$r$ symmetric tensors are both path-connected if $r$ is not more than the complex generic rank; these results also extend to border rank and symmetric border rank over $\\mathbb{C}$. Over $\\mathbb{R}$, the set of rank-$r$ tensors is path-connected if it has the expected dimension but the corresponding result for symmetric ra"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1804.08060","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"}