{"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/the-polynomial-set-associated-with-a-fixed","title":"The Polynomial Set Associated with a Fixed Number of Matrix-Matrix Multiplications","arxiv_id":"2504.01500","date":"2025-04-02","proceeding":null,"authors":["Elias Jarlebring","Gustaf Lorentzon"],"abstract":"We consider the problem of computing matrix polynomials $p(X)$, where $X$ is a large dense matrix, with as few matrix-matrix multiplications as possible. More precisely, let $\\Pi_{2^{m}}^*$ represent the set of polynomials computable with $m$ matrix-matrix multiplications, but with an arbitrary number of matrix additions and scaling operations. We characterize this set through a tabular parameterization. By deriving equivalence transformations of the tabular representation, we establish new methods that can be used to construct elements of $\\Pi_{2^{m}}^*$ and determine general properties of the set. The transformations allow us to eliminate variables and prove that the dimension is bounded by $m^2$, which is subsequently proven to be sharp, i.e., $\\dim(\\Pi_{2^m}^*)=m^2$. Consequently, we have identified a parameterization that, to the best of our knowledge, is the first minimal parameterization. We also conduct a study using computational tools from algebraic geometry to determine the largest degree $d$ such that all polynomials of that degree belong to $\\Pi_{2^{m}}^*$, or its closure. In many cases, the computational setup is constructive in the sense that it can also be used to determine a specific evaluation scheme for a given polynomial.","url_abs":"https://arxiv.org/abs/2504.01500v2","url_pdf":"https://arxiv.org/pdf/2504.01500v2.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":"the-polynomial-set-associated-with-a-fixed","repo_url":"https://github.com/gustaflorentzon/polynomial-set-paper","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}