Liouvillian first integrals of differential equations: A Galoisian approach
Abstract
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.
Related
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BibTeX
@misc{kumbhakar2026liouvillian,
title = {{Liouvillian first integrals of differential equations: A Galoisian approach}},
author = {Partha Kumbhakar and Chitrarekha Sahu and Varadharaj R. Srinivasan},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.08623},
eprint = {2610.08623},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.08623},
}
RIS
TY - UNPB
TI - Liouvillian first integrals of differential equations: A Galoisian approach
AU - Partha Kumbhakar
AU - Chitrarekha Sahu
AU - Varadharaj R. Srinivasan
PY - 2026
DA - 2026/10/06
PB - arXiv
DO - 10.48550/arxiv.2610.08623
UR - https://arxiv.org/abs/2610.08623
AB - 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.
ER -