{"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/linear-system-of-hypersurfaces-passing","title":"Linear system of hypersurfaces passing through a Galois orbit","arxiv_id":"2310.10361","date":"2023-10-16","proceeding":null,"authors":["Shamil Asgarli","Dragos Ghioca","Zinovy Reichstein"],"abstract":"Let $d$ and $n$ be positive integers, and $E/F$ be a separable field extension of degree $m=\\binom{n+d}{n}$. We show that if $|F| > 2$, then there exists a point $P\\in \\mathbb{P}^n(E)$ which does not lie on any degree $d$ hypersurface defined over $F$. In other words, the $m$ Galois conjugates of $P$ impose independent conditions on the $m$-dimensional $F$-vector space of degree $d$ forms in $x_0, x_1, \\ldots, x_n$. As an application, we determine the maximal dimensions of linear systems $\\mathcal{L}_1$ and $\\mathcal{L}_2$ of hypersurfaces in $\\mathbb P^n$ over a finite field $F$, where every $F$-member of $\\mathcal{L}_1$ is reducible and every $F$-member of $\\mathcal{L}_2$ is irreducible.","url_abs":"https://arxiv.org/abs/2310.10361v3","url_pdf":"https://arxiv.org/pdf/2310.10361v3.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":"linear-system-of-hypersurfaces-passing","repo_url":"https://github.com/sasgarli/hypersurfaces-Galois-orbit","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}