Unified Analytic Solution of Polynomial Equations in Exterior Algebras: A Comprehensive Extension with Complete Mathematical Foundations

Unified Analytic Solution of Polynomial Equations in Exterior Algebras: A Comprehensive Extension with Complete Mathematical Foundations | 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 Search Search ADVANCED SEARCH SCROLL This is a preprint and has not been peer reviewed.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. 20 October 2025 V1 Latest version Share on Unified Analytic Solution of Polynomial Equations in Exterior Algebras: A Comprehensive Extension with Complete Mathematical Foundations Authors : Dongqi Liu 0009-0006-4018-9292 and shifa liu 0009-0003-6570-2812 [email protected] Authors Info & Affiliations 191 views 107 downloads Contents Abstract Supplementary Material Information & Authors Metrics & Citations View Options References Figures Tables Media Share Abstract 本文提出了一个全面的理论和计算框架，用于求解外部（格拉斯曼）代数中的多元多项式方程组。这些代数的反交换性质对经典求解方法提出了根本性挑战。我们通过引入严格定义的外部路径排序算子 PE 系统地应对这些挑战，该算子投影到完全反对称的子空间上，同时保留基本的代数结构。我们通过显式有限阶段过程构造了一个外代数闭包有限元，并证明在精确的泛化条件下，它包含外部多项式方程组任意系统的显式解。我们工作的核心是一个统一的解析解公式，用外部临界点、外部临界值和明确定义的外部多项式 Ψj 来表示根，这些多项式结合了组合系数和路径排序校正，这些校正是通过排列群的完全反对称和得出的。求解方法为根提供了显式解析表达式，采用统一形式： X k = X （n−1） + n−1 j=1 PE Ψj（Y） 1/n ω jk n ， 0 ≤ k ≤ n − 1， （1） 其中 X （n−1） 是外部临界点，Y = （y （0）,. .., y （n−2）） 是外部临界值，ωn = e 2πi/n。我们提供具有完全数学严谨性的完整建设性证明，包括外部牛顿恒等式的详细推导和收敛分析。开发的算法利用外部代数的无能性特性，通过全面的误差分析实现了 O（n 2 m 3 2 2m） 复杂度。数值验证证明了各种测试用例中 10 −14 残差的精度，包括高维和条件不佳的系统。这项工作表明，在反交换设置中结合微分和几何运算的外部代数结构中存在显式解析解。应用非常广泛，涵盖超对称物理、微分几何和拓扑场论。此外，我们建立了一个完整的外加罗瓦理论，将经典可解性标准扩展到反交换设置，并证明了阿贝尔-鲁菲尼定理的外部版本，它阐明了我们的一般解析解与基本外代数中仅由根数表示的解之间的区别。 Supplementary Material File (exterior_algebra1.pdf) Download 492.60 KB Information & Authors Information Version history V1 Version 1 20 October 2025 DOI 10.22541/au.176099575.50945437/v1 Copyright This work is licensed under a Creative Commons Attribution 4.0 International License Keywords algebraic closure anti-commutative algebra complete mathematical foundations differential geometry explicit solution exterior algebra galois theory geometric computation grassmann algebra multivariate polynomial systems path-ordering supersymmetry 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 107 downloads .FvxKWukQNSOunydq8rnd { width: 100px; } Citations Download citation Dongqi Liu, shifa liu. Unified Analytic Solution of Polynomial Equations in Exterior Algebras: A Comprehensive Extension with Complete Mathematical Foundations. Authorea . 20 October 2025. DOI: https://doi.org/10.22541/au.176099575.50945437/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 . 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