{"pk":64977,"title":"Generalized polynomials and hyperplane functions in \\((\\mathbb{Z}/p^k\\mathbb{Z})^n\\)","subtitle":null,"abstract":"For \\(p\\) prime, let \\(\\mathcal{H}^n\\) be the linear span of indicator functions of hyperplanes in \\((\\mathbb{Z}/p^k\\mathbb{Z})^n\\). We establish new upper bounds on the dimension of \\(\\mathcal{H}^n\\) over \\(\\mathbb{Z}/p\\mathbb{Z}\\), or equivalently, on the rank of point-hyperplane incidence matrices in \\((\\mathbb{Z}/p^k\\mathbb{Z})^n\\) over \\(\\mathbb{Z}/p\\mathbb{Z}\\). Our proof is based on a variant of the polynomial method using binomial coefficients in \\(\\mathbb{Z}/p^k\\mathbb{Z}\\) as generalized polynomials. We also establish additional necessary conditions for a function on \\((\\mathbb{Z}/p^k\\mathbb{Z})^n\\) to be an element of \\(\\mathcal{H}^n\\).\n \nMathematics Subject Classifications: 05B20, 05B25, 05A10\n \nKeywords: Hyperplanes, generalized polynomials, binomial coefficients","language":"en","license":{"name":"Creative Commons Attribution 4.0","short_name":"CC BY 4.0","text":"Attribution — You must give appropriate credit, provide a link to the license, and indicate if changes were made. You may do so in any reasonable manner, but not in any way that suggests the licensor endorses you or your use.\n\nNo additional restrictions — You may not apply legal terms or technological measures that legally restrict others from doing anything the license permits.","url":"https://creativecommons.org/licenses/by/4.0"},"keywords":[{"word":"Hyperplanes"},{"word":"generalized polynomials"},{"word":"binomial coefficients"}],"section":"Research Articles","is_remote":true,"remote_url":"https://escholarship.org/uc/item/2c331052","frozenauthors":[{"first_name":"Izabella","middle_name":"","last_name":"Łaba","name_suffix":"","institution":"Department of Mathematics, UBC, Vancouver, B.C. V6T 1Z2, Canada","department":""},{"first_name":"Charlotte","middle_name":"","last_name":"Trainor","name_suffix":"","institution":"Department of Mathematics, UBC, Vancouver, B.C. V6T 1Z2, Canada Current address: Department of Mathematics, Harvard University, Cambridge, MA 02138, U.S.A.","department":""}],"date_submitted":"2025-03-14T17:10:20Z","date_accepted":"2025-03-14T17:10:20Z","date_published":"2025-03-15T07:00:00Z","render_galley":null,"galleys":[{"label":"","type":"pdf","path":"https://journalpub.escholarship.org/combinatorial_theory/article/64977/galley/49787/download/"}]}