发表机构
Ben-Gurion University of the Negev; Indian Institute of Science Education and Research (IISER) Mohali（内盖夫本-古里安大学; 印度科学教育研究所（IISER）莫哈利分校）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
利用微分Galois理论证明一阶微分方程组存在Liouvillian首次积分等价于其在Liouvillian Picard-Vessiot扩张或环中存在，给出Singer定理的新证明，并展示Liouvillian假设不可去掉。
AI 中文摘要
利用微分Galois理论——具体而言，即Magid的完全Picard-Vessiot闭包理论——我们证明：一个n个变量的一阶微分方程组存在Liouvillian首次积分，当且仅当它在一个Liouvillian Picard-Vessiot扩张中存在首次积分，当且仅当它在其Picard-Vessiot环中存在首次积分。因此，具有Liouvillian首次积分的方程组，其首次积分存在于一个从有理函数域出发，先经过有限代数扩张，再添加一个积分的指数，最后添加一个积分而得到的微分域中。这给出了Singer关于自治平面系统Liouvillian首次积分定理（Trans. Amer. Math. Soc. 333, 1992）及其由Aziz等人（arXiv:2512.15522）最近推广到n个变量自治系统的新证明。我们的结果在特征为零且常数域代数闭的任意微分域上成立；特别地，它们适用于非自治系统。最后，我们展示了一个在非Liouvillian Picard-Vessiot扩张中存在首次积分，但在其Picard-Vessiot环中不存在首次积分的系统，从而证明Liouvillian假设不能被去掉。
英文摘要
Using differential Galois theory---specifically, Magid's theory of the complete Picard-Vessiot closure---we show that a system of first-order differential equations in $n$ variables admits a Liouvillian first integral if and only if it admits a first integral in a Liouvillian Picard-Vessiot extension, if and only if it admits a first integral in its Picard-Vessiot ring. Consequently, a system with a Liouvillian first integral admits one in a differential field obtained from the field of rational functions by first taking a finite algebraic extension, then adjoining an exponential of an integral, and then an integral. This yields a new proof of Singer's theorem on Liouvillian first integrals of autonomous planar systems (Trans.\ Amer.\ Math.\ Soc.\ 333, 1992) and of its recent extension to autonomous systems in $n$ variables by Aziz et al. (arXiv:2512.15522). Our results hold over any differential field of characteristic zero with an algebraically closed field of constants; in particular, they apply to non-autonomous systems. Finally, we exhibit a system that admits a first integral in a non-Liouvillian Picard-Vessiot extension but none in its Picard-Vessiot ring and thus establishing that the Liouvillian hypothesis cannot be dropped.
Comments21 pages. Comments are welcome
