{"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/asymptotic-behaviour-of-integer-programming","title":"Asymptotic behaviour of integer programming and the $\\text{v}$-function of a graded filtration","arxiv_id":"2403.08435","date":"2024-03-13","proceeding":null,"authors":["Antonino Ficarra","Emanuele Sgroi"],"abstract":"The $\\text{v}$-function of a graded filtration $\\mathcal{I}=\\{I_{[k]}\\}_{k\\ge0}$ is introduced. Under the assumption that $\\mathcal{I}$ is Noetherian, we prove that the $\\text{v}$-function $\\text{v}(I_{[k]})$ is an eventually quasi-linear function. This result applies to several situations, including ordinary powers, and integral closures of ordinary powers, among others. As another application, we investigate the asymptotic behaviour of certain integer programming problems. Finally, we present the \\textit{Macaulay2} package $\\texttt{VNumber}$.","url_abs":"https://arxiv.org/abs/2403.08435v7","url_pdf":"https://arxiv.org/pdf/2403.08435v7.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":"asymptotic-behaviour-of-integer-programming","repo_url":"https://github.com/emanuelesgroi/vnumber","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}