{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/depth-first-search-for-tensor-rank-and-border","title":"Depth-first search for tensor rank and border rank over finite fields","arxiv_id":"2411.14676","date":"2024-11-22","proceeding":null,"authors":["Jason Yang"],"abstract":"We present an $O^*\\left(|\\mathbb{F}|^{(R-n_*)\\left(\\sum_d n_d\\right)+n_*}\\right)$-time algorithm for determining whether a tensor of shape $n_0\\times\\dots\\times n_{D-1}$ over a finite field $\\mathbb{F}$ has rank $\\le R$, where $n_*:=\\max_d n_d$; we assume without loss of generality that $\\forall d:n_d\\le R$. We also extend this problem to its border rank analog, i.e., determining tensor rank over rings of the form $\\mathbb{F}[x]/(x^H)$, and give an $O^*\\left(|\\mathbb{F}|^{H\\sum_{1\\le r\\le R} \\sum_d \\min(r,n_d)}\\right)$-time algorithm. Both of our algorithms use polynomial space.","url_abs":"https://arxiv.org/abs/2411.14676v1","url_pdf":"https://arxiv.org/pdf/2411.14676v1.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"links_only","authors_date_abstract":"arXiv metadata, CC0 1.0 (https://info.arxiv.org/help/license), from the Kaggle arXiv metadata snapshot of 2026-09-12"},"code_links":[{"paper_slug":"depth-first-search-for-tensor-rank-and-border","repo_url":"https://github.com/coolcomputery/tensor-cpd-search","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":1,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}