{"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/on-the-number-of-mathbb-f-q-zeros-of-families","title":"$\\mathbb{F}_q$-zeros of sparse trivariate polynomials and toric 3-fold codes","arxiv_id":"2105.10071","date":"2021-05-21","proceeding":null,"authors":["Kyle Meyer","Ivan Soprunov","Jenya Soprunova"],"abstract":"For a given lattice polytope $P$ in $\\mathbb{R}^3$, consider the space $\\mathcal{L}_P$ of trivariate polynomials over a finite field $\\mathbb{F}_q$, whose Newton polytopes are contained in $P$. We give an upper bound for the maximum number of $\\mathbb{F}_q$-zeros of polynomials in $\\mathcal{L}_P$ in terms of the Minkowski length of $P$ and $q$, the size of the field. Consequently, this produces lower bounds for the minimum distance of toric codes defined by evaluating elements of $\\mathcal{L}_P$ at the points of the algebraic torus $(\\mathbb{F}_q^*)^3$. Our approach is based on understanding factorizations of polynomials in $\\mathcal{L}_P$ with the largest possible number of non-unit factors. The related combinatorial result that we obtain is a description of Minkowski sums of lattice polytopes contained in $P$ with the largest possible number of non-trivial summands.","url_abs":"https://arxiv.org/abs/2105.10071v2","url_pdf":"https://arxiv.org/pdf/2105.10071v2.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":"on-the-number-of-mathbb-f-q-zeros-of-families","repo_url":"https://github.com/isoprou/minkowski-length","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}