{"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/computing-the-binomial-part-of-a-polynomial","title":"Computing the Binomial Part of a Polynomial Ideal","arxiv_id":"2307.09394","date":"2023-07-18","proceeding":null,"authors":["Martin Kreuzer","Florian Walsh"],"abstract":"Given an ideal $I$ in a polynomial ring $K[x_1,\\dots,x_n]$ over a field $K$, we present a complete algorithm to compute the binomial part of $I$, i.e., the subideal ${\\rm Bin}(I)$ of $I$ generated by all monomials and binomials in $I$. This is achieved step-by-step. First we collect and extend several algorithms for computing exponent lattices in different kinds of fields. Then we generalize them to compute exponent lattices of units in 0-dimensional $K$-algebras, where we have to generalize the computation of the separable part of an algebra to non-perfect fields in characteristic $p$. Next we examine the computation of unit lattices in affine $K$-algebras, as well as their associated characters and lattice ideals. This allows us to calculate ${\\rm Bin}(I)$ when $I$ is saturated with respect to the indeterminates by reducing the task to the 0-dimensional case. Finally, we treat the computation of ${\\rm Bin}(I)$ for general ideals by computing their cellular decomposition and dealing with finitely many special ideals called $(s,t)$-binomial parts. All algorithms have been implemented in SageMath.","url_abs":"https://arxiv.org/abs/2307.09394v1","url_pdf":"https://arxiv.org/pdf/2307.09394v1.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":"computing-the-binomial-part-of-a-polynomial","repo_url":"https://github.com/abacus42/binomial-part","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":1,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}