Unified Analytic Solution of Polynomial Equations in Vector Algebraic Closure

Unified Analytic Solution of Polynomial Equations in Vector Algebraic Closure | 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 Search Search ADVANCED SEARCH SCROLL This is a preprint and has not been peer reviewed.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. 6 October 2025 V1 Latest version Share on Unified Analytic Solution of Polynomial Equations in Vector Algebraic Closure Authors : Dongqi Liu and shifa liu 0009-0003-6570-2812 Authors Info & Affiliations https://doi.org/10.22541/au.175977864.44766141/v1 177 views 124 downloads Contents Abstract Supplementary Material Information & Authors Metrics & Citations View Options References Figures Tables Media Share Abstract This paper extends the differential algebraic framework for solving polynomial equations to the vector algebraic setting. We establish a rigor ous theoretical foundation for solving systems of multivariate polynomial equations and matrix eigenvalue problems within a vector algebraic clo sure V. The solution methodology provides explicit analytical expressions for roots of multivariate polynomial systems and eigenvalues of matrices, taking the unified form: n−1 xk =x(n−1) + j=1 Φj(Y)1/nωj(k−1) n, 0 ≤k≤n−1, where x(n−1) is the multivariate critical point, Y = (y(0),...,y(n−2)) are vector critical values, Φj ∈ Q(A)[Y] are explicit vector polynomials with combinatorial correction terms, and ωn = e2πi/n. We provide com plete constructive proofs, derive combinatorial expressions for the cor rection coefficients γ(n) j, and present detailed algorithms with complexity analysis. Extensive numerical validation demonstrates machine-precision a ccuracy across various test cases, including ill-conditioned multivariate systems and challenging eigenvalue problems. This work reconciles with classical impossibility results while demonstrating that explicit analytic solutions exist in appropriately extended algebraic structures. Supplementary Material File (v3.pdf) Download 568.78 KB Information & Authors Information Version history V1 Version 1 06 October 2025 Copyright This work is licensed under a Creative Commons Attribution 4.0 International License Keywords combinatorial correction eigenvalue problems explicit solution multivariate polynomial systems numerical computation vector algebraic closure Authors Affiliations Dongqi Liu The University of North Carolina at Chapel Hill View all articles by this author shifa liu 0009-0003-6570-2812 View all articles by this author Metrics & Citations Metrics Article Usage 177 views 124 downloads .FvxKWukQNSOunydq8rnd { width: 100px; } Citations Download citation Dongqi Liu, shifa liu. Unified Analytic Solution of Polynomial Equations in Vector Algebraic Closure. Authorea . 06 October 2025. DOI: https://doi.org/10.22541/au.175977864.44766141/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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