Multivariate Extension of Exterior Algebra Polynomial Solving: A Comprehensive Theoretical and Computational Framework

Multivariate Extension of Exterior Algebra Polynomial Solving: A Comprehensive Theoretical and Computational Framework | 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 Search Search ADVANCED SEARCH SCROLL This is a preprint and has not been peer reviewed.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 Multivariate Extension of Exterior Algebra Polynomial Solving: A Comprehensive Theoretical and Computational Framework Authors : Dongqi Liu 0009-0006-4018-9292 and shifa liu 0009-0003-6570-2812 [email protected] Authors Info & Affiliations 160 views 125 downloads Contents Abstract Supplementary Material Information & Authors Metrics & Citations View Options References Figures Tables Media Share Abstract 本文提出了外部代数多项式求解框架在多元系统的全面扩展。在已建立的单变量外代数闭包理论[1]的基础上，我们开发了一种用于求解外代数中多元多项式方程组的分层结构。这些代数的反交换性质提出了基本挑战，我们通过引入多变量外部路径排序算子和递归闭包结构来解决这些挑战。我们为多元外代数提供严格的数学基础，包括完整的基描述、维度公式和具有明确符号约定的产品结构。我们证明，在精确的泛型条件下，任意多元外部多项式系统的所有解都可以通过涉及多元临界点、临界值张量和显式定义的外部多项式的显式解析公式来表示。求解方法为根提供了显式的分析表达式，详细的复杂度分析显示了 O（n p p j=1 m 3 j 2 2mj） 最坏情况的计算复杂度。广泛的数值验证证明了各种测试用例（包括双变量二次系统和三变量三次系统）的 10 −14 残差的精度。这些算法通过稀疏张量表示和分层变量处理实现了实际效率，有效复杂度通常明显低于结构化问题的最坏情况限制。此外，我们建立了完整的多元外伽罗瓦理论，将经典可解性准则扩展到多元反交换设置，并证明基本定理的多元外版本。这项工作表明，显式解析解存在于多元外部代数结构中，这些结构在反交换环境中结合了微分和几何运算，同时尊重多元外部 Abel-Ruffini 定理施加的限制。 Supplementary Material File (multiple exterior algebra.pdf) Download 436.16 KB Information & Authors Information Version history V1 Version 1 20 October 2025 DOI 10.22541/au.176099581.13550706/v1 Copyright This work is licensed under a Creative Commons Attribution 4.0 International License Keywords algebraic closure anti-commutative algebra critical value tensor explicit solution exterior algebra galois theory grassmann algebra hierarchical construction multivariate polynomial systems pathordering 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 160 views 125 downloads .FvxKWukQNSOunydq8rnd { width: 100px; } Citations Download citation Dongqi Liu, shifa liu. Multivariate Extension of Exterior Algebra Polynomial Solving: A Comprehensive Theoretical and Computational Framework. Authorea . 20 October 2025. DOI: https://doi.org/10.22541/au.176099581.13550706/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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