{"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/minimization-of-hypersurfaces","title":"Minimization of hypersurfaces","arxiv_id":"2110.04625","date":"2021-10-09","proceeding":null,"authors":["Andreas-Stephan Elsenhans","Michael Stoll"],"abstract":"Let $F \\in \\mathbb{Z}[x_0, \\ldots, x_n]$ be homogeneous of degree $d$ and assume that $F$ is not a `nullform', i.e., there is an invariant $I$ of forms of degree $d$ in $n+1$ variables such that $I(F) \\neq 0$. Equivalently, $F$ is semistable in the sense of Geometric Invariant Theory. Minimizing $F$ at a prime $p$ means to produce $T \\in \\operatorname{Mat}(n+1, \\mathbb{Z}) \\cap \\operatorname{GL}(n+1, \\mathbb{Q})$ and $e \\in \\mathbb{Z}_{\\ge 0}$ such that $F_1 = p^{-e} F([x_0, \\ldots, x_n] \\cdot T)$ has integral coefficients and $v_p(I(F_1))$ is minimal among all such $F_1$. Following Koll\\'ar, the minimization process can be described in terms of applying weight vectors $w \\in \\mathbb{Z}_{\\ge 0}^{n+1}$ to $F$. We show that for any dimension $n$ and degree $d$, there is a complete set of weight vectors consisting of $[0,w_1,w_2,\\dots,w_n]$ with $0 \\le w_1 \\le w_2 \\le \\dots \\le w_n \\le 2 n d^{n-1}$. When $n = 2$, we improve the bound to $d$. This answers a question raised by Koll\\'ar. These results are valid in a more general context, replacing $\\mathbb{Z}$ and $p$ by a PID $R$ and a prime element of $R$. Based on this result and a further study of the minimization process in the planar case $n = 2$, we devise an efficient minimization algorithm for ternary forms (equivalently, plane curves) of arbitrary degree $d$. We also describe a similar algorithm that allows to minimize (and reduce) cubic surfaces. The algorithms are available in the computer algebra system Magma.","url_abs":"https://arxiv.org/abs/2110.04625v3","url_pdf":"https://arxiv.org/pdf/2110.04625v3.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":"minimization-of-hypersurfaces","repo_url":"https://github.com/michaelstollbayreuth/weights","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}