{"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/on-the-maximal-number-of-columns-of-a-d","title":"On the Maximal Number of Columns of a $Δ$-modular Integer Matrix: Bounds and Computations","arxiv_id":"2111.06294","date":"2021-11-11","proceeding":null,"authors":["Gennadiy Averkov","Matthias Schymura"],"abstract":"We study the maximal number of pairwise distinct columns in a $\\Delta$-modular integer matrix with $m$ rows. Recent results by Lee et al. provide an asymptotically tight upper bound of $O(m^2)$ for fixed $\\Delta$. We complement this and obtain an upper bound of the form $O(\\Delta)$ for fixed $m$, and with the implied constant depending polynomially on $m$.","url_abs":"https://arxiv.org/abs/2111.06294v2","url_pdf":"https://arxiv.org/pdf/2111.06294v2.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":"on-the-maximal-number-of-columns-of-a-d","repo_url":"https://github.com/mschymura/delta-classification","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":1,"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}