Unified Analytic Solution of Polynomial Equations in Commutative Banach Algebras via Differential Algebraic Methods

Unified Analytic Solution of Polynomial Equations in Commutative Banach Algebras via Differential Algebraic Methods | Authorea try { document.documentElement.classList.add('js'); } catch (e) { } var _gaq = _gaq || []; _gaq.push(['_setAccount', 'G-8VDV14Y67G']); _gaq.push(['_trackPageview']); (function() { var ga = document.createElement('script'); ga.type = 'text/javascript'; ga.async = true; ga.src = ('https:' == document.location.protocol ? 'https://ssl' : 'http://www') + '.google-analytics.com/ga.js'; var s = document.getElementsByTagName('script')[0]; s.parentNode.insertBefore(ga, s); })(); Skip to main content Preprints Collections Wiley Open Research IET Open Research Ecological Society of Japan All Collections About About Authorea FAQs Contact Us Quick Search anywhere Search for preprint articles, keywords, etc. Search Search ADVANCED SEARCH SCROLL This is a preprint and has not been peer reviewed. Data may be preliminary. 15 October 2025 V1 Latest version Share on Unified Analytic Solution of Polynomial Equations in Commutative Banach Algebras via Differential Algebraic Methods Authors : Dongqi Liu 0009-0006-4018-9292 and shifa liu 0009-0003-6570-2812 [email protected] Authors Info & Affiliations https://doi.org/10.22541/au.176055038.85881877/v1 121 views 142 downloads Contents Abstract Supplementary Material Information & Authors Metrics & Citations View Options References Figures Tables Media Share Abstract This paper establishes a rigorous framework for solving polynomial equations in commutative unital Banach algebras by extending the differential algebraic closure approach. We prove that all solutions of a degree-n polynomial equation P (x) = 0 in a commutative unital Banach algebra A can be analytically expressed within a Banach differential algebraic closure KA. We provide complete constructive proofs with detailed combinatorial analysis, derive explicit expressions for the correction coefficients γ(n) m,and present a detailed algorithm with complexity analysis. The work reconciles with the Abel-Ruffini theorem by demonstrating that while solutions in radicals are impossible for general quintic and higher-degree equations, explicit analytic solutions exist in the appropriately extended Banach differential algebraic closure KA. Extensive validation through special cases and error analysis confirms the method’s correctness and numerical stability. New contributions include enhanced combinatorial interpretations, improved asymptotic analysis, practical implementation strategies for handling spectral constraints, and detailed spectral estimation methods for infinite-dimensional cases. Supplementary Material File (banach1 (1).pdf) Download 526.11 KB Information & Authors Information Version history V1 Version 1 15 October 2025 Copyright This work is licensed under a Creative Commons Attribution 4.0 International License Keywords banach algebra combinatorial analysis differential algebraic closure operator theory polynomial equations spectral theory Authors Affiliations Dongqi Liu 0009-0006-4018-9292 View all articles by this author shifa liu 0009-0003-6570-2812 [email protected] View all articles by this author Metrics & Citations Metrics Article Usage 121 views 142 downloads .FvxKWukQNSOunydq8rnd { width: 100px; } Citations Download citation Dongqi Liu, shifa liu. Unified Analytic Solution of Polynomial Equations in Commutative Banach Algebras via Differential Algebraic Methods. Authorea . 15 October 2025. DOI: https://doi.org/10.22541/au.176055038.85881877/v1 If you have the appropriate software installed, you can download article citation data to the citation manager of your choice. Simply select your manager software from the list below and click Download. For more information or tips please see 'Downloading to a citation manager' in the Help menu . Format Please select one from the list RIS (ProCite, Reference Manager) EndNote BibTex Medlars RefWorks Direct import Tips for downloading citations document.getElementById('citMgrHelpLink').addEventListener('click', function() { popupHelp(this.href); return false; }); $(".js__slcInclude").on("change", function(e){ if ($(this).val() == 'refworks') $('#direct').prop("checked", false); $('#direct').prop("disabled", ($(this).val() == 'refworks')); }); View Options View options PDF View PDF Figures Tables Media Share Share Share article link Copy Link Copied! Copying failed. 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