{"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/computations-associated-with-the-resonance","title":"Computations associated with the resonance arrangement","arxiv_id":"2106.09940","date":"2021-06-18","proceeding":null,"authors":["Zachary Chroman","Mihir Singhal"],"abstract":"The resonance arrangement $\\mathcal{A}_n$ is the arrangement of hyperplanes in $\\mathbb{R}^n$ given by all hyperplanes of the form $\\sum_{i \\in I} x_i = 0$, where $I$ is a nonempty subset of $\\{1,\\dots,n\\}$. We consider the characteristic polynomial $\\chi(\\mathcal{A}_n; t)$ of the resonance arrangement, whose value $R_n$ at $-1$ is of particular interest, and corresponds to counts of generalized retarded functions in quantum field theory, among other things. No formula is known for either the characteristic polynomial or $R_n$, though $R_n$ has been computed up to $n=8$. By exploiting symmetry and using computational methods, we compute the characteristic polynomial of $\\mathcal{A}_9$, and thus obtain $R_9$. The coefficients of the characteristic polynomial are also equal to the so-called Betti numbers of the complexified hyperplane arrangement; that is, the coefficient of $t^{n-i}$ is denoted by the Betti number $b_i(\\mathcal{A}_n)$. Explicit formulas are known for the Betti numbers up to $b_3(\\mathcal{A}_n)$. Using computational methods, we also obtain an explicit formula for $b_4(\\mathcal{A}_n)$, which gives the $t^{n-4}$ coefficient of the characteristic polynomial.","url_abs":"https://arxiv.org/abs/2106.09940v2","url_pdf":"https://arxiv.org/pdf/2106.09940v2.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":"computations-associated-with-the-resonance","repo_url":"https://github.com/mihirsinghal/resonance-arrangement","is_official":1,"mentioned_in_paper":1,"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}