Roots in the semiring of finite deterministic dynamical systems

Roots in the semiring of finite deterministic dynamical systems | Research Square window.SnipcartSettings = { analytics: { enabled: false } }; (function() { var accessVector = localStorage.getItem('access_vector') || ''; window.dataLayer = window.dataLayer || []; if (accessVector) { window.dataLayer.push({ user: { profile: { profileInfo: { snid: accessVector } } } }); } })(); (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0],j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src='https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-K279D39R'); Browse Preprints In Review Journals COVID-19 Preprints AJE Video Bytes Research Tools Research Promotion AJE Professional Editing AJE Rubriq About Preprint Platform In Review Editorial Policies Our Team Advisory Board Help Center Sign In Submit a Preprint Cite Share Download PDF Research Article Roots in the semiring of finite deterministic dynamical systems François Doré, Kévin Perrot, Antonio E. Porreca, Sara Riva, Marius Rolland This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-6278096/v1 This work is licensed under a CC BY 4.0 License Status: Under Review Version 1 posted 11 You are reading this latest preprint version Abstract Finite discrete-time dynamical systems (FDDS) model phenomena that evolve deterministically in discrete time. It is possible to define sum and product operations on these systems (disjoint union and direct product, respectively) giving a commutative semiring. This algebraic structure led to several works employing polynomial equations to model hypotheses on phenomena modelled using FDDS.To solve these equations, algorithms for performing the division and computing $k$-th roots are needed.In this paper, we propose two polynomial algorithms for these tasks, under the condition that the result is a connected FDDS. This ultimately leads to an efficient solution to equations of the type $AX^k=B$ for connected $X$ and some generalisations.These results are some of the important final steps for solving more general polynomial equations on FDDS. discrete dynamical systems root of graph direct product Full Text Additional Declarations No competing interests reported. Cite Share Download PDF Status: Under Review Version 1 posted Editorial decision: Revision requested 15 Sep, 2025 Reviews received at journal 12 Sep, 2025 Reviews received at journal 12 Aug, 2025 Reviews received at journal 04 Aug, 2025 Reviewers agreed at journal 09 May, 2025 Reviewers agreed at journal 06 May, 2025 Reviewers agreed at journal 06 May, 2025 Reviewers invited by journal 06 May, 2025 Editor assigned by journal 24 Mar, 2025 Submission checks completed at journal 22 Mar, 2025 First submitted to journal 21 Mar, 2025 You are reading this latest preprint version Research Square lets you share your work early, gain feedback from the community, and start making changes to your manuscript prior to peer review in a journal. As a division of Research Square Company, we’re committed to making research communication faster, fairer, and more useful. We do this by developing innovative software and high quality services for the global research community. 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Also discoverable on Platform About Our Team In Review Editorial Policies Advisory Board Help Center Resources Author Services Accessibility API Access RSS feed Manage Cookie Preferences © Research Square 2026 | ISSN 2693-5015 (online) Privacy Policy Terms of Service Do Not Sell My Personal Information {"props":{"pageProps":{"initialData":{"identity":"rs-6278096","acceptedTermsAndConditions":true,"allowDirectSubmit":false,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":452751602,"identity":"b6996d8c-217d-45a7-b931-7831c5d17ca4","order_by":0,"name":"François Doré","email":"","orcid":"","institution":"University of Bordeaux","correspondingAuthor":false,"prefix":"","firstName":"François","middleName":"","lastName":"Doré","suffix":""},{"id":452751603,"identity":"c43323b3-bf6e-42cb-88f2-ce13d36b4efc","order_by":1,"name":"Kévin Perrot","email":"","orcid":"","institution":"Aix-Marseille University","correspondingAuthor":false,"prefix":"","firstName":"Kévin","middleName":"","lastName":"Perrot","suffix":""},{"id":452751604,"identity":"a94d29bd-d1fb-421c-84e8-4cb85b257dad","order_by":2,"name":"Antonio E. 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