A Unified Solution Formula for Polynomial Equations in Noncommutative Operator Algebras via Differential-Combinatorial Methods

A Unified Solution Formula for Polynomial Equations in Noncommutative Operator Algebras via Differential-Combinatorial 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 Search Search ADVANCED SEARCH SCROLL This is a preprint and has not been peer reviewed.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. 8 October 2025 V1 Latest version Share on A Unified Solution Formula for Polynomial Equations in Noncommutative Operator Algebras via Differential-Combinatorial 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.175994530.00057479/v1 191 views 128 downloads Contents Abstract Supplementary Material Information & Authors Metrics & Citations View Options References Figures Tables Media Share Abstract This paper establishes a comprehensive framework for solving degree-n polynomial equations P(x) = 0 in unital noncommutative operator alge bras, including C*-algebras, von Neumann algebras, and general operator systems. We construct a noncommutative operator differential-algebraic closure that extends the coefficient algebra by adjoining critical values of P, all roots of unity, and is closed under the extraction of p-th roots via holomorphic functional calculus and under solving polynomial equations with ordered noncommutative coefficients. The central result is an explicit solution formula expressing all roots of P(x) = 0 within this closure. This formula generalizes classical radical solutions to the noncommutative operator setting through the introduc tion of novel noncommutative combinatorial correction terms Γ(n) m, rigor ously derived from the noncommutative Faa di Bruno formula adapted for operator-valued functions. We provide complete proofs of the fundamental relations, detailed ver ifications for degrees 2 through 6, a practical algorithm with complexity analysis, and demonstrate consistency with the noncommutative Abel- Ruffini theorem. The framework is shown to be robust under weakened commutativity assumptions and includes a comprehensive treatment of the holomorphic functional calculus for the n-th root operation in opera tor algebras, including spectral avoidance conditions and domain consid erations for unbounded operators. New contributions include scalable combinatorial computation meth ods, detailed numerical implementation strategies for operator algebras, quantitative spectral management analysis, and extensions to free proba bility and noncommutative geometry settings with explicit error bounds for operator-valued computations. Supplementary Material File (noncommutative_operator1.pdf) Download 454.62 KB Information & Authors Information Version history V1 Version 1 08 October 2025 Copyright This work is licensed under a Creative Commons Attribution 4.0 International License Keywords abel-ruffini theorem c*-algebra combinatorial correction differential algebraic closure free derivative holomorphic functional calculus noncommutative operator algebra operator-valued functional calculus polynomial equations von neumann algebra 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 191 views 128 downloads .FvxKWukQNSOunydq8rnd { width: 100px; } Citations Download citation Dongqi Liu, shifa liu. A Unified Solution Formula for Polynomial Equations in Noncommutative Operator Algebras via Differential-Combinatorial Methods. Authorea . 08 October 2025. DOI: https://doi.org/10.22541/au.175994530.00057479/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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