{"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/ideals-of-spaces-of-degenerate-matrices","title":"Ideals of Spaces of Degenerate Matrices","arxiv_id":"2106.00735","date":"2021-06-01","proceeding":null,"authors":["Julian Vill","Mateusz Michałek","Alexander Taveira Blomenhofer"],"abstract":"The variety $ \\mathrm{Sing}_{n, m} $ consists of all tuples $ X = (X_1,\\ldots, X_m) $ of $ n\\times n $ matrices such that every linear combination of $ X_1,\\ldots, X_m $ is singular. Equivalently, $X\\in\\mathrm{Sing}_{n,m}$ if and only if $\\det(\\lambda_1 X_1 + \\ldots + \\lambda_m X_m) = 0$ for all $ \\lambda_1,\\ldots, \\lambda_m\\in \\mathbb Q $. Makam and Wigderson asked whether the ideal generated by these equations is always radical, that is, if any polynomial identity that is valid on $ \\mathrm{Sing}_{n, m} $ lies in the ideal generated by the polynomials $\\det(\\lambda_1 X_1 + \\ldots + \\lambda_m X_m)$. We answer this question in the negative by determining the vanishing ideal of $ \\mathrm{Sing}_{2, m} $ for all $ m\\in \\mathbb N $. Our results exhibit that there are additional equations arising from the tensor structure of $ X $. More generally, for any $ n $ and $ m\\ge n^2 - n + 1 $, we prove there are equations vanishing on $ \\mathrm{Sing}_{n, m} $ that are not in the ideal generated by polynomials of type $\\det(\\lambda_1 X_1 + \\ldots + \\lambda_m X_m)$. Our methods are based on classical results about Fano schemes, representation theory and Gr\\\"obner bases.","url_abs":"https://arxiv.org/abs/2106.00735v2","url_pdf":"https://arxiv.org/pdf/2106.00735v2.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":"ideals-of-spaces-of-degenerate-matrices","repo_url":"https://github.com/a44l/ideals-of-spaces-of-degenerate-matrices","is_official":1,"mentioned_in_paper":0,"mentioned_in_github":0,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"syntology_url":null,"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}