# Overview of publicly available degradation data sets for tasks within
prognostics and health management ††thanks: Citation: Hagmeyer, S.; Mauthe,
F.; Zeiler, P. (2021): Creation of Publicly Available Data Sets for
Prognostics and Diagnostics Addressing Data Scenarios Relevant to Industrial
Applications. International Journal of Prognostics and Health Management,
Volume 12, Issue 2, DOI: 10.36001/ijphm.2021.v12i2.3087.

Fabian Mauthe
Institute for Technical Reliability and Prognostics IZP, Esslingen University
of Applied Sciences
Goeppingen, Germany
<EMAIL_ADDRESS>
Christopher Braun
Institute of Industrial Manufacturing and Management IFF, University of
Stuttgart
Stuttgart, Germany
<EMAIL_ADDRESS>
Julian Raible
Institute of Industrial Manufacturing and Management IFF, University of
Stuttgart
Stuttgart, Germany
<EMAIL_ADDRESS>
Peter Zeiler
Institute for Technical Reliability and Prognostics IZP, Esslingen University
of Applied Sciences
Goeppingen, Germany
<EMAIL_ADDRESS>
Marco F. Huber
Institute of Industrial Manufacturing and Management IFF, University of
Stuttgart
Fraunhofer Institute of Manufacturing Engineering and Automation IPA
Stuttgart, Germany
<EMAIL_ADDRESS>

##  1 Prognostics and Health Management

In the realm of modern manufacturing, significant emphasis is placed on
improving the reliability, performance, and service life of complex
engineering systems, which has catalyzed the emergence of Prognostics and
Health Management (PHM) as an essential discipline in the past two decades [1,
2]. PHM includes a range of methods dedicated to the early detection,
prognosis, and mitigation of impending faults or failures in ESs, thereby
facilitating sophisticated maintenance strategies and enhancing operational
efficiency [3].

Central to the efficacy of PHM methods is the acquisition and analysis of
degradation data, which encapsulates the evolving health condition of ESs over
time [3]. Degradation data serves as a rich source of information, offering
invaluable insights into the underlying degradation processes, failure modes,
and performance trends of ESs [2, 4]. Through the integration of advanced
sensing technologies, data analytics, and machine learning algorithms,
researchers and practitioners can harness degradation data to develop
prognostic models, optimize maintenance schedules, and make informed
operational decisions [1].

In the remainder of this paper, the tasks of diagnostics and prognostics are
listed, and an overview of publicly available degradation data sets is
presented as well as briefly described. The authors aim to update the overview
regularly in the future, as new data sets are continually being published in
the dynamic research field of PHM.

##  2 Tasks of Diagnostics and Prognostics

Extensive and representative degradation data enables addressing a specific
task—prognosis—within PHM, with the subdivision of PHM into distinct tasks.
Based on [1], [5], and [4], the tasks of diagnostics and prognostics within
PHM can be subdivided as follows:

  * •

Fault detection/anomaly detection: Detect a fault state or anomaly in an ES
without considering its root cause. This results in a binary classification
problem with the states fault or no fault.

  * •

Diagnosis: Assign one or more causes to a detected fault state.

  * •

Health assessment: Assess the state of degradation or the current risk of
failure of an ES based on its current condition.

  * •

Prognosis: Prediction of the future degradation behavior or the current
Remaining Useful Life (RUL).

The prognosis task poses the greatest challenge in diagnostics and prognostics
within PHM [6]. This difficulty stems from the intricate nature of failure
mechanisms and the multitude of factors influencing degradation progression.
Therefore, sufficient degradation data is of huge importance for this task and
needs to be comprehensively available.

##  3 Data Sets for Diagnostics and Prognostics

Table Appendix: Overview of Publicly Available Data Sets in the _Appendix:
Overview of Publicly Available Data Sets_ contains an overview of publicly
available data sets for the tasks of diagnostics and prognostics within PHM,
sorted alphabetically. This is the extended overview from [4], which already
contains 70 data sets.

The designation of the data sets is divided into two parts, consisting of the
source or platform and the title under which they were published. In addition,
the overview contains the following columns:

  * •

Domain and Application: The associated domain from which the data set
originates and the respective application within this domain.

  * •

Task: The corresponding task within PHM addressable with this data set.

  * •

URL: The respective hyperlink to the source or platform hosting the data set.

##  4 Description of Data Sets

The overview of publicly available degradation data sets in Table Appendix:
Overview of Publicly Available Data Sets comprises a total of 79 data sets.
These data sets originate from eleven different domains. The domains as well
as the applications presented do not claim to be exhaustive and are based
solely on the available data sets. With 17 assigned data sets, the _Mechanical
component_ domain is the most extensive, followed by the _Electrical
component_ domain with 13 assigned data sets. The two most frequently
occurring applications, _Bearing_ (ten) and _Battery_ (six), also originate
from these two domains. In total, the domains contain 36 different
applications.

Except for the _Building_ domain, which is sparsely represented in PHM, the
data sets are evenly distributed for the remaining domains, with at least four
and at most nine assignments per domain. With only one data set for each
application, the _Mechatronic system_ domain contains the greatest variation
of applications. Moreover, there are five data sets with unknown applications
and therefore, no domain can be assigned. Data sets with degradation data
resulting from a simulated application are assigned to the corresponding
domains but marked. With 32 assigned data sets for prognosis and 29 for
diagnosis, these tasks are the most represented. For fault detection, 15 data
sets are assigned, and for health assessment, 3 data sets are assigned. The
_Electrical component_ domain contains the most data sets (ten) for the task
prognosis, followed by the _Drive technology_ domain with six and the
_Manufacturing process_ as well as _Process technology_ domains with four
assigned data sets each.

Of the 17 assigned data sets in the _Mechanical component_ domain, only two
are for task prognosis. However, for the task diagnosis, the _Mechanical
component_ domain contains the most data sets with eleven assignments.
Followed by the _Mechatronic system_ domain with five and the _Robotic_ domain
with four assigned data sets. Regarding fault detection, the domains
_Mechanical component_ and _Production system_ contain the most data sets with
four assignments each. With two data sets with an _Unknown_ domain and one
data set from the _Drive technology_ domain, the overview contains the fewest
data sets for the health assessment task.

## References

  * [1] Enrico Zio.  Prognostics and health management (phm): Where are we and where do we (need to) go in theory and practice.  Reliability Engineering & System Safety, 218:108119, 2022.
  * [2] Yaguo Lei, Naipeng Li, Liang Guo, Ningbo Li, Tao Yan, and Jing Lin.  Machinery health prognostics: A systematic review from data acquisition to rul prediction.  Mechanical Systems and Signal Processing, 104:799–834, 2018.
  * [3] Vepa Atamuradov, Kamal Medjaher, Pierre Dersin, Benjamin Lamoureux, and Noureddine Zerhouni.  Prognostics and health management for maintenance practitioners - review, implementation and tools evaluation.  International Journal of Prognostics and Health Management, 8(3), 2017.
  * [4] Simon Hagmeyer, Fabian Mauthe, and Peter Zeiler.  Creation of publicly available data sets for prognostics and diagnostics addressing data scenarios relevant to industrial applications.  International Journal of Prognostics and Health Management, 12(2), 2021.
  * [5] Xiaodong Jia, Bin Huang, Jianshe Feng, Haoshu Cai, and Jay Lee.  Review of phm data competitions from 2008 to 2017.  Annual Conference of the PHM Society, 10(1), 2018.
  * [6] Mojtaba Kordestani, Mehrdad Saif, Marcos E. Orchard, Roozbeh Razavi-Far, and Khashayar Khorasani.  Failure prognosis and applications—a survey of recent literature.  IEEE Transactions on Reliability, 70(2):728–748, 2021.

## Appendix: Overview of Publicly Available Data Sets

Overview of publicly available data sets |  |  |  |
---|---|---|---|---
No. |  Data Set Designation
Reference to Data Origin |  Domain
Application |  Task |  URL
Table 0 – continued from previous page |  |  |  |
No. |  Data Set Designation
Reference to Data Origin |  Domain
Application |  Task |  URL
Continued on next page |  |  |  |
1 |  AIDAR Lab - Air Compressor
Nishchal K. Verma, R. K. Sevakula, S. Dixit and Salour A. (2016). Intelligent Condition Based Monitoring using Acoustic Signals for Air Compressors, IEEE Transactions on Reliability, vol. 65, no. 1, pp. 291-309. |  Mechatronic system
Air Compressor |  Diagnosis |  Link
2 |  AIDAR Lab - Drill Bit
Nishchal K. Verma, R. K. Sevakula, S. Dixit and Salour A. (2015). Data Driven Approach for Drill Bit Monitoring, Reliability Digest, pp. 19-26. |  Manufacturing process
Drill bit |  Diagnosis |  Link
3 |  Aramis - Data Challenge ESREL2020PSAM15
Francesco Cannarile (Aramis Srl, Italy), Michele Compare (Aramis Srl, Italy and Politecnico di Milano, Italy), Piero Baraldi (Politecnico di Milano, Italy), Zhe Yang (Politecnico di Milano, Italy), Enrico Zio (Politecnico di Milano, Italy and MINES ParisTech, France). |  Unknown
Simulation (time-continuous stochastic process)* |  Fault detection |  Link
4 |  Backblaze - Hard Drive Stats
Backblaze, Inc. |  Drive technology
Disk Drives |  Fault detection |  Link
5 |  Bearing Data Center - Fan and Bearing
Case Western Reserve University, USA |  Mechanical component
Bearing |  Diagnosis |  Link
6 |  GitHub - Predictive Maintenance using PySpark
GitHub User: linya9191 |  Simulation
Simulation* |  Fault detection |  Link
7 |  Kaggle - Air pressure system failures in Scania trucks
Tony Lindgren and Jonas Biteus, Scania CV AB - Stockholm |  Mechatronic system
Air Pressure System |  Fault detection |  Link
8 |  Kaggle - CNC Mill Tool Wear
System-level Manufacturing and Automation Research Testbed (SMART) at the University of Michigan |  Manufacturing process
Mill Tool |  Fault detection |  Link
9 |  Kaggle - Condition Data with Random Recording Time
Hagmeyer, S., Mauthe, F., Zeiler, P. (2021). Creation of Publicly Available Data Sets for Prognostics and Diagnostics Addressing Data Scenarios Relevant to Industrial Applications. International Journal of Prognostics and Health Management, Volume 12, Issue 2, DOI: 10.36001/ijphm.2021.v12i2.3087 |  Process technology
Filtration |  Prognosis |  Link
10 |  Kaggle - E-coating ultrafiltration maintenance
Tinsley Bridge Limited, Sheffield, United Kingdom |  Manufacturing process
Electrophoresis Painting |  Prognosis |  Link
11 |  Kaggle - Genesis Demonstrator
Institut für industrielle Informationstechnik (inIT) der Technischen Hochschule Ostwestfalen-Lippe (IMPROVE) |  Robotic
Pick-and-Place Demonstrator |  Diagnosis |  Link
12 |  Kaggle - Microsoft Azure Predictive Maintenance
Azure AI Notebooks for Predictive Maintenance |  Unknown
Unknown Machine |  Fault detection |  Link
13 |  Kaggle - One Year Industrial Component Degradation
Institut für industrielle Informationstechnik (inIT) der Technischen Hochschule Ostwestfalen-Lippe (IMPROVE) |  Production system
Shrink-Wrapper |  Fault detection |  Link
14 |  Kaggle - Kaggle - Predictive Maintenance 1
Creators Yuan Yao and Zhao Yuqi |  Production system
Log Data |  Fault detection |  Link
15 |  Kaggle - Preventive to Predictive Maintenance
Hagmeyer, S., Mauthe, F., Zeiler, P. (2021). Creation of Publicly Available Data Sets for Prognostics and Diagnostics Addressing Data Scenarios Relevant to Industrial Applications. International Journal of Prognostics and Health Management, Volume 12, Issue 2, DOI: 10.36001/ijphm.2021.v12i2.3087 |  Process technology
Filtration |  Prognosis |  Link
16 |  Kaggle - Production Plant Data for Condition Monitoring
von Birgelen, Alexander; Buratti, Davide; Mager, Jens; Niggemann, Oliver (2018). Self-Organizing Maps for Anomaly Localization and Predictive Maintenance in Cyber-Physical Production Systems. Proceedings of the CIRP Conference on Manufacturing Systems (CIRP CMS 2018) CIRP-CMS |  Production system
Productionline |  Fault detection |  Link
17 |  Kaggle - Prognosis based on Varying Data Quality
Hagmeyer, S., Mauthe, F., Zeiler, P. (2021). Creation of Publicly Available Data Sets for Prognostics and Diagnostics Addressing Data Scenarios Relevant to Industrial Applications. International Journal of Prognostics and Health Management, Volume 12, Issue 2, DOI: 10.36001/ijphm.2021.v12i2.3087 |  Process technology
Filtration |  Prognosis |  Link
18 |  Kaggle - Pump
Kaggle User: UnknownClass |  Process technology
Water Pump |  Fault detection |  Link
19 |  Kaggle - Sensor Fault Detection
Schneider-Electric, France |  Electrical component
Temperature Sensor |  Fault detection |  Link
20 |  Kaggle - Versatile Production System
Institut für industrielle Informationstechnik (inIT) der Technischen Hochschule Ostwestfalen-Lippe (IMPROVE) |  Production system
Versatile Production System |  Fault detection |  Link
21 |  Mendeley - Brushless DC motor
Mazzoleni, Mirko; Scandella, Matteo; Previdi, Fabio; Pispola, Giulio (2019). Data for: First endurance activity of a Brushless DC motor for aerospace applications - REPRISE project, Mendeley Data, V2, doi: 10.17632/m58 bdhy2df.2 |  Drive technology
Brushless DC Motor |  Prognosis |  Link
22 |  Mendeley - Diesel Engine Faults
Denys Pestana-Viana - Federal Center of Technological Education Celso Suckow da Fonseca (CEFET-RJ), Rio de Janeiro, Brazil |  Drive technology
Diesel Engine |  Diagnosis |  Link
23 |  Mendeley - HUST Bearing
Hong, Hoang Si; Thuan, Nguyen (2023), “HUST bearing: a practical dataset for ball bearing fault diagnosis”, Mendeley Data, V3, doi: 10.17632/cbv7jyx4p9.3 |  Mechanical component
Bearing |  Diagnosis |  Link
24 |  MFPT - Condition Based Maintenance Fault
Data Assembled and Prepared on behalf of MFPT by Dr Eric Bechhoefer, Chief Engineer, NRG Systems |  Mechanical component
Bearing |  Diagnosis |  Link
25 |  NASA - Accelerated Battery Life Testing
Fricke, K., Nascimento, R., Corbetta, M., Kulkarni, C., Viana, F. (2023). Prognosis of Li-ion Batteries Under Large Load Variations Using Hybrid Physics-Informed Neural Networks. Proccedings of the Annual Conference of the PHM Society, 15(1). https://doi.org/10.36001/phmconf.2023.v15i1.3463 |  Electrical component
Battery |  Prognosis |  Link
26 |  NASA - Bearing Data Set
J. Lee, H. Qiu, G. Yu, J. Lin, and Rexnord Technical Services. IMS, University of Cincinnati. "Bearing Data Set", NASA Ames Prognostics Data Repository, NASA Ames Research Center, Moffett Field, CA |  Mechanical component
Bearing |  Prognosis |  Link
27 |  NASA - Capacitor Electrical Stress
J. Renwick, C. Kulkarni, and J Celaya "Capacitor Electrical Stress Data Set", NASA Ames Prognostics Data Repository, NASA Ames Research Center, Moffett Field, CA |  Electrical component
Electrolytic Capacitors |  Prognosis |  Link
28 |  NASA - Capacitor Electrical Stress 2
J. Celaya, C. Kulkarni, G. Biswas, and K. Goebel "Capacitor Electrical Stress Data Set - 2", NASA Ames Prognostics Data Repository, NASA Ames Research Center, Moffett Field, CA |  Electrical component
Electrolytic Capacitors |  Prognosis |  Link
29 |  NASA - CFRP Composites Data Set
Abhinav Saxena, Kai Goebel, Cecilia C. Larrosa, and Fu-Kuo Chang "CFRP Composites Data Set", NASA Ames Prognostics Data Repository, NASA Ames Research Center, Moffett Field, CA |  Materials
Carbon Fibre-Reinforced Polymer Composite |  Prognosis |  Link
30 |  NASA - HIRF Battery
C. Kulkarni, E. Hogge, C. Quach and K. Goebel "HIRF Battery Data Set", NASA Ames Prognostics Data Repository (http://ti.arc.nasa.gov/project/prognostic-data-repository), NASA Ames Research Center, Moffett Field, CA |  Electrical component
Battery |  Prognosis |  Link
31 |  NASA - IGBT
J. Celaya, Phil Wysocki, and K. Goebel "IGBT Accelerated Aging Data Set", NASA Ames Prognostics Data Repository, NASA Ames Research Center, Moffett Field, CA |  Electrical component
Transistor |  Prognosis |  Link
32 |  NASA - Li-ion Battery Aging Datasets
B. Saha and K. Goebel. "Battery Data Set", NASA Ames Prognostics Data Repository, NASA Ames Research Center, Moffett Field, CA |  Electrical component
Battery |  Prognosis |  Link
33 |  NASA - Milling Data set
A. Agogino and K. Goebel. BEST lab, UC Berkeley. "Milling Data Set ", NASA Ames Prognostics Data Repository, NASA Ames Research Center, Moffett Field, CA |  Manufacturing process
Milling |  Progrognosis |  Link
34 |  NASA - MOSFET Thermal Overstress Aging
J. R. Celaya, A. Saxena, S. Saha, and K. Goebel "MOSFET Thermal Overstress Aging Data Set", NASA Ames Prognostics Data Repository, NASA Ames Research Center, Moffett Field, CA |  Electrical component
Transistor |  Prognosis |  Link
35 |  NASA - Randomized Battery Usage Data Set
B. Bole, C. Kulkarni, and M. Daigle "Randomized Battery Usage Data Set", NASA Ames Prognostics Data Repository, NASA Ames Research Center, Moffett Field, CA |  Electrical component
Battery |  Prognosis |  Link
36 |  NASA - Small Satellite Power Simulation
C. Kulkarni and A. Guarneros "Small Satellite Power Simulation Data Set", NASA Ames Prognostics Data Repository, NASA Ames Research Center, Moffett Field, CA |  Electrical component
Battery |  Diagnosis |  Link
37 |  NASA - Turbofan engine degradation simulation data set
A. Saxena and K. Goebel. "Turbofan Engine Degradation Simulation Data Set", NASA Ames Prognostics Data Repository, NASA Ames Research Center, Moffett Field, CA |  Drive technology
Aircraft Engine* |  Prognosis |  Link
38 |  NASA - Turbofan engine degradation simulation data set 2
M. Chao, C.Kulkarni, K. Goebel and O. Fink (2021). "Aircraft Engine Run-to-Failure Dataset under real flight conditions", NASA Ames Prognostics Data Repository, NASA Ames Research Center, Moffett Field, CA |  Drive technology
Aircraft Engine* |  Prognosis |  Link
39 |  NIST - Robot Arm Position Accuracy
National Institute of Standards and Technology NIST (USA), Engineering Laboratory, Intelligent Systems Division |  Robotic
Industrial Robot |  Diagnosis |  Link
40 |  OpenEI - Evaluation of Building Fault
Lin Guanjing and Robin Mitchel, Lawrence Berkeley National Laboratory, (2019) |  Building
Building |  Diagnosis |  Link
41 |  OpenEI - Gearbox Fault Diagnosis
Yogesh Pandy, National Renewable Energy Laboratory |  Mechanical component
Gear |  Fault detection |  Link
42 |  PHM Data Challenge 2008 - Turbofan
NASA Prognostics Center of Excellence |  Drive technology
Aircraft Engine* |  Prognosis |  Link
43 |  PHM Data Challenge 2009 - Gearbox Fault Detection
PHM Society, Gearbox fault detection data set, 2010 |  Mechanical component
Gear System (Gears, Bearings, Shafts) |  Diagnosis |  Link
44 |  PHM Data Challenge 2010 - CNC milling machine cutters
X. Li, B.S. Lim, J.H. Zhou, S. Huang, S.J. Phua, K.C. Shaw, and M.J. Er; Singapore Institute of Manufacturing Technology, 71 Nanyang Drive, Singapore 638075 School of Electrical and Electronic Engineering, Nanyang Technological University, Nanyang Avenue, Singapore 639798 |  Manufacturing process
CNC Milling Machine Cutters |  Prognosis |  Link
45 |  PHM Data Challenge 2011 - Anemometer Fault Detection
Creators Unknown |  Mechanical component
Anemometer |  Fault detection |  Link
46 |  PHM Data Challenge 2013 - Unknown
Creators Unknown |  Unknown
Unknown |  Health assessment |  Link
47 |  PHM Data Challenge 2014 - Unknown
Creators Unknown |  Unknown
Unknown |  Health assessment |  Link
48 |  PHM Data Challenge 2015 - Plant Fault Detection
Creators Unknown |  Production system
Industrial Plant Monitoring |  Diagnosis |  Link
49 |  PHM Data Challenge 2016 - Semiconductor CMP
Crystec Technology Trading GmbH |  Manufacturing process
Chemicalmechanical Planarization System |  Prognosis |  Link
50 |  PHM Data Challenge 2017 - Bogie Vehicle
Creators Unknown |  Mechanical component
Bogie |  Diagnosis |  Link
51 |  PHM Data Challenge 2018 - Ion Mill in Waver Manufacturing
Kai Goebel (at this time at NASA) |  Production system
Ion Mill Etching Tool |  Prognosis |  Link
52 |  PHM Data Challenge 2019 - Fatigue Cracks
Creators Unknown |  Materials
Aluminum Structure with Dynamic Loading |  Prognosis |  Link
53 |  PHM Data Challenge 2020 Europe - Filtration System
Eker, O. F., Camci, F., & Jennions, I. K. (2016). Physics-based prognostic modelling of filter clogging phenomena. Mechanical Systems and Signal Processing, 75, pp. 395-412. |  Process technology
Filtration |  Prognosis |  Link
54 |  PHM Data Challenge 2021 - Turbofan 2
M. Chao, C.Kulkarni, K. Goebel and O. Fink (2021). "Aircraft Engine Run-to-Failure Dataset under real flight conditions", NASA Ames Prognostics Data Repository, NASA Ames Research Center, Moffett Field, CA (additional validation data compared to data set NASA - Turbofan engine degradation simulation data set 2) |  Drive technology
Aircraft Engine* |  Prognosis |  Link
55 |  PHM Data Challenge 2021 Europe: SCARA-robot
Swiss Centre for Electronics and Microtechnology (CSEM) |  Robotic
Industrial Robot |  Diagnosis |  Link
56 |  PHM Data Challenge 2022 - Rock Drills
Epiroc Rock Drills AB, Örebro, 702 25, Sweden and Linköping University, Linköping, 581 83, Sweden |  Mechatronic system
Rock Drills |  Diagnosis |  Link
57 |  PHM Data Challenge 2022 Europe - PCB Production line
Danilo Giordano and Martino Trevisan, (2022). "PHME Data Challenge". European conference of the prognostics and health management society. / BITRON: Strada del Portone 95 - 10095 Grugliasco (TO), Italy, https://www.bitron.com/en |  Production system
PCB Production Line |  Diagnosis |  Link
58 |  PHM Data Challenge 2023 - Gearbox
PHM Society, Gearbox data set, 2023 |  Mechanical component
Gear |  Diagnosis |  Link
59 |  PHM Data Challenge 2023 Asia Pacific - Experimental Propulsion System
Japan Aerospace Exploration Agency (JAXA) |  Drive technology
Propulsion System* |  Health assessment |  Link
60 |  PHM IEEE Data Challenge 2012 - FEMTO Bearing Data Set
FEMTO-ST Institute, Besançon, France |  Mechanical component
Bearing |  Prognosis |  Link
61 |  PHM IEEE Data Challenge 2014 - Fuel Cell
FCLAB Research Federation (FR CNRS 3539, France) |  Electrical component
Fuel Cell |  Prognosis |  Link
62 |  PHM IEEE Data Challenge 2023 - Planetary Gearbox
Mälardalen University (MDU) / Mälardalen Industrial Technology Center (MITC) |  Mechanical component
Planetary Gear |  Diagnosis |  Link
63 |  PHM Society - Electromechanical Ball Screw Drive
GE Research and University of Tennessee Knoxville in part by the Advanced Research Projects Agency-Energy (ARPA-E), U.S. Department of Energy, under Award Number DE-AR0001290 |  Mechatronic system
Electromechanical Ball Screw Drive* |  Diagnosis |  Link
64 |  ResearchGate - Driveline Unbalanced Shaft
Giacomo Barbieri and David Sanchez-Londono and Laura Cattaneo and Luca Fumagalli and David Romero |  Mechanical component
Unbalanced shaft (driveline) |  Diagnosis |  Link
65 |  SDOL - Diagnostics 101 bearing data
Kim, S., An, D., Choi, J-H. (2020). Diagnostics 101: A Tutorial for Fault Diagnostics of Rolling Element Bearing Using Envelope Analysis in MATLAB. Applied Sciences. 10(20):7302. doi:10.3390/app10207302 |  Mechanical component
Bearing |  Fault detection |  Link
66 |  SDOL - HS Gear
Creators Unknown |  Mechanical component
Bearing |  Fault detection |  Link
67 |  SDOL - KAU Gear
Creators Unknown |  Mechanical component
Bearing |  Diagnosis |  Link
68 |  Toyota Research Institute - Battery Cycle Life
Severson, K.A., Attia, P.M., Jin, N. et al. Data-driven prediction of battery cycle life before capacity degradation. Nat Energy 4, 383–391 (2019). https://doi.org/10.1038/s41560-019-0356-8 |  Electrical component
Battery |  Prognosis |  Link
69 |  UCI - AI4I 2020 Predictive Maintenance
Matzka, S. (2020). “Explainable Artificial Intelligence for Predictive Maintenance Applications," Proceedings of the Third International Conference on Artificial Intelligence for Industries (AI4I), pp. 69-74, doi: 10.1109/AI4I49448.2020.00023 . |  Production system
Manufacturing Machine |  Diagnosis |  Link
70 |  UCI - Condition Based Maintenance of Naval Propulsion Plants Data Set
1: DIBRIS - University of Genoa 2: School of Marine Science and Technology, Newcastle University |  Drive technology
Naval Propulsion Plant* |  Prognosis |  Link
71 |  UCI - Condition monitoring of hydraulic systems Data Set
ZeMA - Center for Mechatronics and Automation Technology Saarbrücken, Germany |  Mechatronic system
Hydraulic System |  Prognosis |  Link
72 |  UCI - Mechanical Analysis
F. Bergadano, A. Giordana, L. Saitta, (1990). University of Torino, Italy |  Mechatronic system
Electromechanical Devices |  Diagnosis |  Link
73 |  UCI - Robot Execution Failures Data Set
Universidade Nova de Lisboa, Monte da Caparica, Portugal |  Robotic
Industrial Robot |  Diagnosis |  Link
74 |  UCI - Steel Plates Faults
Semeion Research Center of Sciences of Communication - Rome, Italy |  Materials
Steel Plates Faults |  Diagnosis |  Link
75 |  Uni Lulea - Vibration Signals Wind Turbines
Luleå University of Technology: Martin del Campo Barraza, Sergio (Department of Computer Science, Electrical and Space Engineering, Embedded Internet Systems Lab); Sandin, Fredrik (Department of Computer Science, Electrical and Space Engineering, Embedded Internet Systems Lab); Strömbergsson, Daniel (Department of Engineering Sciences and Mathematics, Machine Elements) |  Mechanical component
Bearing (Wind Turbines) |  Fault detection |  Link
76 |  Uni Paderborn KAt - Bearing Damage
University Paderborn Kat |  Mechanical component
Bearing |  Diagnosis |  Link
77 |  Virkler
Virkler, D. A., Hillberry, B. M. and Goel, P. K. (1979). The Statistical Nature of Fatigue Crack Propagation. Journal of Engineering Materials and Technology, vol. 101, 148–153. |  Materials
Aluminum Plate |  Prognosis |  Link
78 |  WASEDA - Fault Detection and Classification
De Bruijn, B., Nguyen, T. A., Bucur, D., & Tei, K. (2016). Benchmark datasets for fault detection and classification in sensor data. In A. Ahrens, O. Postolache, & C. Benavente-Peces (Eds.), SENSORNETS 2016 - Proceedings of the 5th International Confererence on Sensor Networks, pp. 185-195. . SciTePress. https://doi.org/10.5220/0005637901850195 |  Electrical component
Sensors |  Diagnosis |  Link
79 |  Zenodo - Predictive maintenance dataset
Huawei Munich Research Center |  Mechatronic system
Elevator |  Prognosis |  Link
  *[PHM]: Prognostics and Health Management
  *[ESs]: engineering system
  *[ES]: engineering system
  *[RUL]: Remaining Useful Life


Real Liouvillian Extensions of Partial Differential Fields

Real Liouvillian Extensions
of Partial Differential Fields

Teresa CRESPO a, Zbigniew HAJTO b and Rouzbeh MOHSENI b

T. Crespo, Z. Hajto and R. Mohseni

a) Departament de Matemàtiques i Informàtica, Universitat de Barcelona,
a) Gran Via de les Corts Catalanes 585, 08007 Barcelona, Spain
<EMAIL_ADDRESS>http://www.ub.edu/tn/personal/crespo.php

b) Faculty of Mathematics and Computer Science, Jagiellonian University,
a) ul. Łojasiewicza 6, 30-348 Kraków, Poland<EMAIL_ADDRESS><EMAIL_ADDRESS>
Received February 28, 2021, in final form October 25, 2021; Published online
October 29, 2021

In this paper, we establish Galois theory for partial differential systems
defined over formally real differential fields with a real closed field of
constants and over formally $p$-adic differential fields with a $p$-adically
closed field of constants. For an integrable partial differential system
defined over such a field, we prove that there exists a formally real (resp.
formally $p$-adic) Picard–Vessiot extension. Moreover, we obtain a uniqueness
result for this Picard–Vessiot extension. We give an adequate definition of
the Galois differential group and obtain a Galois fundamental theorem in this
setting. We apply the obtained Galois correspondence to characterise formally
real Liouvillian extensions of real partial differential fields with a real
closed field of constants by means of split solvable linear algebraic groups.
We present some examples of real dynamical systems and indicate some
possibilities of further development of algebraic methods in real dynamical
systems.

real Liouvillan extension; real and $p$-adic Picard–Vessiot theory; split
solvable algebraic group; gradient dynamical systems; integrability

12H05; 37J35; 12D15; 14P05

##  1 Introduction

Galois theory for linear differential equations is the differential
counterpart of classical Galois theory. The idea of Galois of characterising
those polynomial equations solvable by radicals by means of the group of
permutations of the roots which preserve the relations between them was
paralleled in the work of Picard and Vessiot who characterised linear
differential equations solvable by quadratures by means of the group of linear
automorphisms of the vector space of solutions that preserve the differential
relations between them. In a similar way as the work of Galois was later
formalized by Artin, the one of Picard and Vessiot was formalized by using
differential algebra, more precisely introducing the notion of a differential
field $K$ to be a field endowed with a derivation. The constants of $K$ are
defined as the elements on which the derivation vanishes and form a subfield
$\mathcal{C}$ of $K$. A satisfactory Galois theory for linear differential
equations defined over a differential field $K$ with algebraically closed
field of constants $\mathcal{C}$ was established by Kolchin, under the name of
Picard–Vessiot theory (see [17]). For such a differential equation, Kolchin
proved the existence and uniqueness up to $K$-differential isomorphism of a
Picard–Vessiot field, the analog of the splitting field in classical Galois
theory. The differential Galois group is defined as the group of differential
automorphisms of the Picard–Vessiot field which fix the base field $K$ and it
has the structure of a linear algebraic group defined over the field of
constants $\mathcal{C}$. The fundamental theorem of Picard–Vessiot theory
establishes a bijective correspondence between intermediate differential
fields of the Picard–Vessiot extension and the closed subgroups of the
differential Galois group. We note that the hypothesis that the field of
constants $\mathcal{C}$ of $K$ is algebraically closed was essential in the
work of Kolchin. His results can be extended to integrable partial
differential systems defined over a partial differential field $K$ with
algebraically closed field of constants (see [24, Appendix D]).

In [9], T. Crespo, Z. Hajto and M. van der Put show that the condition that
the field of constants is algebraically closed may be relaxed. They consider
formally real and formally $p$-adic fields, whose definition is given in
Section 2. For a homogeneous linear differential equation defined over a
formally real (resp. formally $p$-adic) ordinary differential field $K$ with a
real closed (resp. $p$-adically closed) field of constants $\mathcal{C}_{K}$,
they prove the existence of a formally real (resp. formally $p$-adic)
Picard–Vessiot extension. Moreover, they obtain a result of uniqueness of the
formally real (resp. formally $p$-adic) Picard–Vessiot extension up to
$K$-differential isomorphism. In model theoretic language, formally real
(resp. formally $p$-adic) differential fields with a real closed (resp.
$p$-adically closed) field of constants are instances of differential fields
$K$ such that the field of constants $\mathcal{C}_{K}$ is existentially closed
in $K$. Under this hypothesis, the existence of a Picard–Vessiot extension is
proved in [12, Theorem 2.2]. In [15] the results in [9] are generalised to the
case when $\mathcal{C}_{K}$ is existentially closed in $K$, large and bounded.

The standard example of a real closed field is the field $\mathbb{R}$ of real
numbers whereas the field of real rational functions in one or several
variables is an example of a formally real field. Picard–Vessiot theory for
formally real differential fields with real closed field of constants allows
then to characterise some aspects of the behaviour of real functions. In [7]
T. Crespo and Z. Hajto characterised real Liouvillian extensions of ordinary
differential fields in terms of real Picard–Vessiot theory. Their result
answers an earlier question of A. Khovanskii, namely “Is it true that a
necessary and sufficient condition for solvability of a real differential
equation by real Liouville functions follows from real Picard–Vessiot
theory?”, which originated in the theory developed in [11]. For a
generalisation and further development of this theory one can consult [16].

In this paper, we establish Galois theory for partial differential systems
over formally real or formally $p$-adic partial differential fields. We prove
the existence of a formally real (resp. formally $p$-adic) Picard–Vessiot
extension for a partial differential system defined over a formally real
(resp. formally $p$-adic) partial differential field $K$ with a real closed
(resp. $p$-adically closed) field of constants. We establish as well a
uniqueness result of the formally real (resp. formally $p$-adic)
Picard–Vessiot extension up to $K$-differential isomorphism. We give an
adequate definition of the differential Galois group in this setting and prove
a Galois correspondence theorem. Due to the fact that real Liouville functions
over the field $\mathbb{R}$ of real numbers have very interesting topological
properties (see [11, 16]), we restrict to formally real fields in our study of
Liouvillianity questions. We characterise formally real Liouvillian extensions
for formally real partial differential fields by means of the differential
Galois group. Previously, we recall in Section 2 the concepts of formally real
field, real closed field, formally $p$-adic field and $p$-adically closed
field as well as the main definitions and known results of Picard–Vessiot
theory. It is worth noting that the algebraic characterisation of real
Liouville functions allows us to expect an algebraic version of the theory
developed by Gel’fond and Khovanskii in [11, 16]. Similar ideas already appear
in Grothendieck’s “Esquisse d’un programme” (see [13, p. 272]), where he
proposed to consider constant functions with values in the real closure
$\overline{\mathbb{Q}}^{r}$ of the field of rational numbers $\mathbb{Q}$ in
the study of integration of Pfaff systems. In the last section of the paper we
present some questions related to the integrability of real dynamical systems
showing the interest of this further development.

##  2 Preliminaries

###  2.1 Ordered fields, formally real fields

In this section we recall the concept of formally real field and real closed
field. More details on these topics may be found in [1, 22].

######  Definition 2.1.

An _ordering_ on a field $k$ is a total order relation $\leq$ on $k$
satisfying, for any $x,y,z\in k$,

  1. $(i)$

$x\leq y\Rightarrow x+z\leq y+z$,

  2. $(ii)$

$0\leq x$ and $0\leq y\Rightarrow 0\leq xy$.

An _ordered field_ $(k,\leq)$ is a field $k$ equipped with an ordering $\leq$.

######  Remark 2.2 ([1, Example 1.1.2]).

Given an ordered field $(k,\leq)$, there is exactly one ordering on the field
of rational functions $k(t)$, extending the one on $k$, such that $t$ is
positive and smaller than any positive element in $k$. If
$P(t)=a_{n}t^{n}+a_{n-1}t^{n-1}+\dots+a_{m}t^{m}\in k[t]$, with $a_{m}\neq 0$,
then $P(t)>0\Leftrightarrow a_{m}>0$. If $P(t)/Q(t)\in k(t)$, then
$P(t)/Q(t)>0\Leftrightarrow P(t)Q(t)>0$.

######  Theorem 2.3 ([1, Theorem 1.1.8]).

Let $k$ be a field. The following properties are equivalent:

  1. $(i)$

$k$ can be ordered,

  2. $(ii)$

$-1$ is not a sum of squares in $k$,

  3. $(iii)$

for $x_{1},\dots,x_{n}\in k$, $\sum_{i=1}^{n}x_{i}^{2}=0\Rightarrow
x_{1}=\dots=x_{n}=0$.

######  Definition 2.4.

A field satisfying the properties of the preceding theorem is called a
_formally real field_.

We note that a formally real field always has characteristic $0$.

######  Definition 2.5.

If $k$ is a formally real field, a _formally real extension_ of $k$ is a field
extension $\ell/k$ such that $\ell$ is a formally real field. A _real closed
field_ is a formally real field that has no nontrivial formally real algebraic
extension.

######  Theorem 2.6 ([1, Theorem 1.2.2]).

Let $k$ be a field. The following properties are equivalent:

  1. $(i)$

$k$ is a real closed field,

  2. $(ii)$

$k$ has a unique ordering,

  3. $(iii)$

the ring $k[i]=k[X]/\big{(}X^{2}+1\big{)}$ is an algebraically closed field.

######  Examples 2.7.

$\mathbb{Q}$, $\mathbb{R}$ are formally real fields, $\mathbb{R}$ is a real
closed field.

######  Definition 2.8.

A _real algebraic closure_ of an ordered field $(k,\leq)$ is an algebraic
field extension $k^{r}$ of $k$ such that $k^{r}$ is a real closed field and
the unique ordering of $k^{r}$ restricts to $\leq$ on $k$.

######  Theorem 2.9 ([22, Theorem 3.10]).

Any ordered field $(k,\leq)$ has a unique $($up to $k$-isomorphism$)$ real
algebraic closure.

###  2.2 Valued fields, $\boldsymbol{p}$-adic fields

In this section we recall the concept of formally $p$-adic field and
$p$-adically closed field. More details on these topics may be found in [23].

######  Definition 2.10.

A _valuation_ of a field $k$ is a map

$\displaystyle v\colon\ k\rightarrow\Gamma\cup\\{\infty\\},$

where $\Gamma$ is a totally ordered abelian group, such that, for all $a$, $b$
in $k$,

  1. $(1)$

$v(a)=\infty\Leftrightarrow a=0$,

  2. $(2)$

$v(ab)=v(a)+v(b)$,

  3. $(3)$

$v(a+b)\geq\min\\{v(a),v(b)\\}$, with equality if $v(a)\neq v(b)$.

We recall that, for $v$ a valuation of a field $k$, the _valuation ring_
$\mathcal{O}$ is defined as $\mathcal{O}:=\\{a\in k\colon v(a)\geq 0\\}$ and
$\mathcal{O}$ has a unique maximal ideal $\mathfrak{m}:=\\{a\in k\colon
v(a)>0\\}$. The _residue field_ is then defined as the quotient
$\mathcal{O}/\mathfrak{m}$.

######  Definition 2.11.

Let $p$ be a prime number. A _$p$ -valuation_ of a field $k$ of characteristic
0 is a valuation $v$ of $k$ such that $v(p)$ is minimal positive in the value
group and the residue field is isomorphic to $\mathbb{Z}/p\mathbb{Z}$.

A _$p$ -valued field_ $(k,v)$ is a characteristic 0 field $k$ equipped with a
$p$-valuation $v$. A _formally $p$-adic field_ is a characteristic 0 field
which can be endowed with a $p$-valuation.

######  Remark 2.12.

If $(k,v)$ is a $p$-valued field, the $p$-valuation $v$ may be extended to a
$p$-valuation of the field of rational functions $k(t)$ (see [23, Example
2.2]).

######  Definition 2.13.

If $k$ is a formally $p$-adic field, a _formally $p$-adic extension_ of $k$ is
a field extension $\ell/k$ such that $\ell$ is a formally $p$-adic field. A
_$p$ -adically closed field_ is a formally $p$-adic field that has no
nontrivial formally $p$-adic algebraic extension.

######  Example 2.14.

$\mathbb{Q}_{p}$ is a $p$-adically closed field.

######  Remark 2.15.

What we call “$p$-adic” is called “$p$-adic of rank one” in [23]. The case of
higher rank $p$-adic fields can be treated in the same way.

######  Definition 2.16.

A _$p$ -adic algebraic closure_ of a $p$-valued field $(k,v)$ is an algebraic
field extension $k^{v}$ of $k$ such that $k^{v}$ is a $p$-adically closed
field and the valuation of $k$ extends to $k^{v}$.

######  Theorem 2.17 ([23, Corollary 3.11]).

Any $p$-valued field $(k,v)$ has a $p$-adic algebraic closure. Let $\ell_{1}$
and $\ell_{2}$ be two $p$-adic closures of $k$. Then $\ell_{1}$ and $\ell_{2}$
are $k$-isomorphic if and only if $k\cap\ell_{1}^{n}=k\cap\ell_{2}^{n}$ for
all $n\in\mathbb{N}$ $($where $\ell_{i}^{n}:=\\{a^{n}|a\in\ell_{i}\\}$,
$i=1,2)$.

###  2.3 Known results on Picard–Vessiot extensions

Let $K$ be a field endowed with a set
$\Delta=\\{\partial_{1},\dots,\partial_{m}\\}$ of pairwise commuting
derivations. Let $\mathcal{C}_{K}$ denote the field of constants of $K$. In
particular, if $m=1$, $K$ is an ordinary differential field. We consider a
differential system

$\displaystyle\partial_{j}Y=A_{j}Y,\qquad 1\leq j\leq m,$ (2.1)

where $A_{j}$ is an $(r\times r)$-matrix with entries in $K$, $1\leq j\leq m$.

A _solution_ for the system (2.1) is a column vector $v$ in the vector space
$L^{r}$, for $L$ some differential field extension of $K$, such that
$\partial_{j}v=A_{j}v$, $1\leq j\leq m$. A _fundamental matrix_ for the system
(2.1) is an $r\times r$ invertible matrix $M$ with entries in some
differential field extension $L$ of $K$ such that $\partial_{j}M=A_{j}M$,
$1\leq j\leq m$. If $M\in\operatorname{GL}_{r}(L)$ is a fundamental matrix for
(2.1), then the set of fundamental matrices for (2.1) with entries in $L$
consists in the matrices of the form $MC$ for
$C\in\operatorname{GL}_{r}(\mathcal{C}_{L})$, for $\mathcal{C}_{L}$ the
constant field of $L$. We note that, if a fundamental matrix exists for (2.1),
then the commutation of the derivatives implies that the matrices $A_{j}$
satisfy

$\displaystyle\partial_{j}A_{i}+A_{i}A_{j}=\partial_{i}A_{j}+A_{j}A_{i},\qquad
1\leq i,j\leq m.$ (2.2)

The relations (2.2) are therefore a necessary condition for the existence of a
common solution to the equations in the system (2.1). We say that the
differential system (2.1) is _integrable_ if the matrices $A_{j}$ satisfy
(2.2).

######  Definition 2.18.

_A Picard–Vessiot extension_ for the differential system (2.1) is a field
extension $L$ of $K$ such that:

  1. $1.$

$L$ is equipped with a set of pairwise commuting derivations extending the
ones in $\Delta$.

  2. $2.$

There exists a fundamental matrix $Z$ for (2.1) with entries in $L$.

  3. $3.$

$L$ is (as a field) generated over K by the entries of $Z$.

  4. $4.$

The field of constants of $L$ is the same as the field of constants of $K$.

When $\mathcal{C}_{K}$ is algebraically closed, a classical result in
differential Galois theory states that a Picard–Vessiot field exists for the
system (2.1) and it is unique up to a $K$-differential isomorphism (see, e.g.,
[5, Theorems 5.6.5 and 5.6.9] for the ordinary case and [24, Appendix D] for
the partial case).

In [9], Crespo, Hajto and van der Put obtained the following theorem on
existence and uniqueness of Picard–Vessiot fields for formally real ordinary
differential fields with real closed field of constants and for formally
$p$-adic ordinary differential fields with $p$-adically closed field of
constants.

######  Theorem 2.19 ([9, Theorem 2]).

Let $K$ be a formally real $($resp. formally $p$-adic$)$ ordinary differential
field with real closed $($resp. $p$-adically closed$)$ field of constants. We
consider a differential system

$\displaystyle Y^{\prime}=AY,$ (2.3)

where $A$ is an $(r\times r)$-matrix with entries in $K$.

  1. $1.$

_Existence_. There exists a formally real $($resp. formally $p$-adic$)$
Picard–Vessiot extension of $K$ for (2.3), i.e., a Picard–Vessiot extension of
$K$ for (2.3) which is also a formally real $($resp. formally $p$-adic$)$
field extension of $K$.

  2. $2.$

_Unicity for the real case_. Let $L_{1}$, $L_{2}$ denote two formally real
Picard–Vessiot extensions of $K$ for (2.3). Suppose that $L_{1}$ and $L_{2}$
have orderings which induce the same ordering on $K$. Then $L_{1}$ and $L_{2}$
are $K$-differentially isomorphic.

  3. $3.$

_Unicity for the_ $\boldsymbol{p}$-_adic case_. Let $L_{1}$, $L_{2}$ denote
two formally $p$-adic Picard–Vessiot extensions of $K$ for (2.3). Suppose that
$L_{1}$ and $L_{2}$ have $p$-adic closures $L_{1}^{+}$ and $L_{2}^{+}$ such
that the $p$-valuations of $L_{1}^{+}$ and $L_{2}^{+}$ induce the same
$p$-valuation on $K$ and such that
$K\cap\big{(}L_{1}^{+}\big{)}^{n}=K\cap\big{(}L_{2}^{+}\big{)}^{n}$ for every
integer $n\geq 2$. Then $L_{1}$ and $L_{2}$ are $K$-differentially isomorphic.

##  3 Galois theory for partial differential systems
over formally real or formally $\boldsymbol{p}$-adic differential fields

In this section we consider an integrable partial differential system (2.1)
defined over a formally real (resp. formally $p$-adic) differential field $K$
with real closed (resp. $p$-adically closed) field of constants
$\mathcal{C}_{K}$. Let $\overline{\mathcal{C}_{K}}$ denote an algebraic
closure of $\mathcal{C}_{K}$. The differential Galois group $G$ for (2.1) is a
$\overline{\mathcal{C}_{K}}/\mathcal{C}_{K}$-form of the differential Galois
group $\overline{G}$ for (2.1) considered as defined over
$\overline{K}:=K\otimes_{\mathcal{C}_{K}}\overline{\mathcal{C}_{K}}$, which is
a partial differential field with algebraically closed field of constants
$\overline{\mathcal{C}_{K}}$. However the differential Galois group $G$ for
(2.1) over $K$ gives more information than $\overline{G}$ on the behaviour of
the solutions to (2.1). In the case when $K$ is a formally real ordinary
differential field and $\mathcal{C}_{K}$ is real closed, we obtained in [7]
that the property for a system $Y^{\prime}=AY$ defined over $K$ to have
solutions which are real Liouville functions is characterised by the
differential Galois group $G$. Therefore it is interesting to extend this
result to partial differential fields.

###  3.1 Picard–Vessiot extensions

In this section, we will use the approach of Kolchin in [18] to show how the
Picard–Vessiot theory of formally real and formally $p$-adic partial
differential fields can be deduced from the ordinary case. Let us note that
Kolchin’s definition of Picard–Vessiot extension for partial differential
fields is different from the one given in Definition 2.18. In [6, Theorem 1]
the equivalence of both definitions is proved.

######  Remark 3.1.

Following Kolchin [18], to a partial differential field $k$ with pairwise
commuting derivations $\partial_{1},\dots,\partial_{m}$ we associate an
ordinary differential field $k_{D}:=k\langle u_{1},\dots,u_{m}\rangle$ with
$u_{1},\dots,u_{m}$ independent differential indeterminates, endowed with the
derivation $D:=u_{1}\partial_{1}+\dots+u_{m}\partial_{m}$. As remarked by
Kolchin, $k_{D}$ and $k$ have the same field of constants. Let us observe
that, as a field, $k_{D}$ is a purely transcendental extension of $k$.
Therefore, if $k$ is a formally real field (resp. a formally $p$-adic field),
then using induction and Remark 2.2 (resp. Remark 2.12), the ordering (resp.
the $p$-valuation) of $k$ may be extended to $k_{D}$, i.e., $k_{D}$ is a
formally real (resp. formally $p$-adic) field.

We consider now a system of the form (2.1) defined over a formally real (resp.
formally $p$-adic) partial differential field. We shall prove the following
theorems on the existence and uniqueness of Picard–Vessiot extensions in this
setting.

######  Theorem 3.2.

  1. $1.$

Let us suppose that $K$ is a formally real partial differential field with
real closed field of constants $\mathcal{C}_{K}$. Then for an integrable
differential system (2.1) defined over $K$, there exists a formally real
differential field $L$ with field of constants $\mathcal{C}_{K}$ such that
$L=K(\\{y_{ij}\\}_{1\leq i,j\leq r})$, for $(y_{ij})_{1\leq i,j\leq
r}\in\operatorname{GL}_{r}(L)$ a fundamental matrix for (2.1), i.e., $L/K$ is
a Picard–Vessiot extension for (2.1).

  2. $2.$

Let us suppose that $K$ is a formally $p$-adic partial differential field with
$p$-adically closed field of constants $\mathcal{C}_{K}$. Then for an
integrable differential system (2.1) defined over $K$, there exists a formally
$p$-adic differential field $L$ with field of constants $\mathcal{C}_{K}$ such
that $L=K(\\{y_{ij}\\}_{1\leq i,j\leq r})$, for $(y_{ij})_{1\leq i,j\leq
r}\in\operatorname{GL}_{r}(L)$ a fundamental matrix for (2.1), i.e., $L/K$ is
a Picard–Vessiot extension for (2.1).

###### Proof.

First we consider an auxiliary differential system

$\displaystyle DY=A_{D}Y,$ (3.1)

where $D=u_{1}\partial_{1}+\dots+u_{m}\partial_{m}$ and
$A_{D}=u_{1}A_{1}+\dots+u_{m}A_{m}\in M_{r\times r}(K_{D})$, with
$u_{1},\dots,u_{m}$ independent differential indeterminates. Since $(K_{D},D)$
is a formally real (resp. formally $p$-adic) ordinary differential field, with
real closed (resp. $p$-adically closed) field of constants $\mathcal{C}_{K}$,
by Theorem 2.19, there exists a formally real (resp. formally $p$-adic)
Picard–Vessiot extension $L_{D}/K_{D}$ for the system (3.1). Let $Z$ denote a
fundamental matrix for (3.1) such that $L_{D}$ is generated over $K_{D}$ by
the entries of $Z$. Let $\overline{\mathcal{C}_{K}}$ denote an algebraic
closure of $\mathcal{C}_{K}$. The derivation $D$ extends uniquely to
$\overline{K}_{D}:=K_{D}\otimes_{\mathcal{C}_{K}}\overline{\mathcal{C}_{K}}$
and
$\overline{L}_{D}:=L_{D}\otimes_{\mathcal{C}_{K}}\overline{\mathcal{C}_{K}}$,
with field of constants $\overline{\mathcal{C}_{K}}$. Since the extensions
$\overline{\mathcal{C}_{K}}/\mathcal{C}_{K}$ and $K_{D}/\mathcal{C}_{K}$ are
linearly disjoint, $\overline{K}_{D}$ and $\overline{L}_{D}$ are fields and
$\overline{L}_{D}$ is generated over $\overline{K}_{D}$ by the entries of $Z$.
Hence $\overline{L}_{D}/\overline{K}_{D}$ is a Picard–Vessiot extension for
(3.1).

On the other hand, since
$\overline{K}:=K\otimes_{\mathcal{C}_{K}}\overline{\mathcal{C}_{K}}$ is a
partial differential field with algebraically closed field of constants
$\overline{\mathcal{C}_{K}}$, we have a Picard–Vessiot extension
$\mathcal{L}/\overline{K}$ for the system (2.1) (see [24, Appendix D]).

The field extensions we have considered up to now are shown in the following
diagram:

---
$\textstyle{\mathcal{L}}$$\textstyle{\overline{L}_{D}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\overline{K}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\overline{K}_{D}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{L_{D}}$$\textstyle{K\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{K_{D}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

We define $\mathcal{L}_{D}:=\mathcal{L}\langle u_{1},\dots,u_{m}\rangle$ and
endow it with the derivation $D=u_{1}\partial_{1}+\dots+u_{m}\partial_{m}$. By
[18, Theorem 1], $\mathcal{L}_{D}$ is a Picard–Vessiot extension of
$\overline{K}_{D}$ for (3.1). By the uniqueness of the Picard–Vessiot
extension in the ordinary case, $\mathcal{L}_{D}$ is isomorphic to
$\overline{L}_{D}$ and we can assume $\mathcal{L}\subset\overline{L}_{D}$.

The Galois group
$G:=\operatorname{Gal}(\overline{\mathcal{C}_{K}}/\mathcal{C}_{K})$ acts on
the field
$\overline{L}_{D}=L_{D}\otimes_{\mathcal{C}_{K}}\overline{\mathcal{C}_{K}}$ by
acting on the second factor. We note that this action commutes with the
derivation $D$ and is continuous. Let $V$ be the
$\overline{\mathcal{C}_{K}}$-vector space of solutions to (2.1) contained in
$\mathcal{L}$. By restricting the action of $G$ on $\overline{L}_{D}$ to $V$
we obtain a continuous semi-linear action. Let $V^{G}:=\\{v\in
V\mid\sigma(v)=v,\,\forall\sigma\in G\\}$. Clearly $V^{G}$ is a
$\mathcal{C}_{K}$-subspace of the $\mathcal{C}_{K}$-vector space $V$. We want
to show that the $\mathcal{C}_{K}$-vector space $V^{G}$ has dimension equal to
the dimension of $V$ over $\overline{\mathcal{C}_{K}}$.

Let $v_{1},\dots,v_{d}$ be a $\mathcal{C}_{K}$-basis of $V^{G}$. We shall
prove first that $v_{1},\dots,v_{d}$ are linearly independent over
$\overline{\mathcal{C}_{K}}$. Assume otherwise and let
$a_{1}v_{1}+\dots+a_{d}v_{d}=0$, with $a_{i}\in\overline{\mathcal{C}_{K}}$,
$1\leq i\leq d$ be a dependence relation with a minimal number of non-zero
coefficients. We may assume $a_{1}\neq 0$ and, by scaling, $a_{1}=1$. By
applying $\sigma\in G$ to the dependence relation and subtracting the obtained
relation from the first one, we obtain from the minimality condition
$\sigma(a_{i})-a_{i}=0$ for $2\leq i\leq d$. Since $\sigma$ is any element in
$G$, we obtain that the coefficients $a_{i}$ belong to $\mathcal{C}_{K}$. This
gives a contradiction, hence $v_{1},\dots,v_{d}$ are
$\overline{\mathcal{C}_{K}}$-linearly independent.

In order to prove that $v_{1},\dots,v_{d}$ generate $V$ over
$\overline{\mathcal{C}_{K}}$, we shall see that a vector $v\in V$ can be
written as a linear combination of vectors in $V^{G}$, with coefficients in
$\overline{\mathcal{C}_{K}}$. In the real case, by Theorem 2.6, we have
$\overline{\mathcal{C}_{K}}=\mathcal{C}_{K}({\rm i})$, where ${\rm
i}=\sqrt{-1}$, and $G=\\{\operatorname{Id},c\\}$ with $c$ defined by $c({\rm
i})=-{\rm i}$. We may then write $v=(1/2)(v+c(v))+{\rm i}\,(1/2{\rm
i})(v-c(v))$ and $(1/2)(v+c(v)),(1/2{\rm i})(v-c(v))$ belong to $V^{G}$. In
the $p$-adic case, $G$ is a pro-finite group. For $v\in
V\subset\overline{L}_{D}=L_{D}\otimes_{\mathcal{C}_{K}}\overline{\mathcal{C}_{K}}$,
there exists a finite extension $\widetilde{\mathcal{C}}/\mathcal{C}$ such
that $v\in L_{D}\otimes_{\mathcal{C}_{K}}\widetilde{\mathcal{C}}$ and we may
assume $\widetilde{\mathcal{C}}/\mathcal{C}_{K}$ Galois. Let
$n:=\big{[}\widetilde{\mathcal{C}}\colon\mathcal{C}_{K}\big{]}$ and
$\widetilde{G}:=\operatorname{Gal}\big{(}\widetilde{\mathcal{C}}/\mathcal{C}_{K}\big{)}=\\{\sigma_{1},\dots,\sigma_{n}\\}$.
By the linear independence of characters and the equality of dimensions, the
map

$\displaystyle\widetilde{\mathcal{C}}\otimes_{\mathcal{C}_{K}}\widetilde{\mathcal{C}}\rightarrow\widetilde{\mathcal{C}}^{n},\qquad
x\otimes y\mapsto(x\sigma_{1}(y),\dots,x\sigma_{n}(y))$

is an isomorphism. We may then find elements
$x_{1},\dots,x_{n},y_{1},\dots,y_{n}$ in $\widetilde{\mathcal{C}}$ such that

$\displaystyle\sum_{i=1}^{n}x_{i}y_{i}=1,\qquad\sum_{i=1}^{n}x_{i}\sigma(y_{i})=0,\qquad\text{for}\quad\sigma\in\widetilde{G},\quad\sigma\neq\operatorname{Id}.$

We have then
$v=\sum_{\sigma\in\widetilde{G}}\sum_{i=1}^{n}x_{i}\sigma(y_{i})\sigma(v)=\sum_{i=1}^{n}x_{i}\big{(}\sum_{\sigma\in\widetilde{G}}\sigma(y_{i}v)\big{)}$
and $\sum_{\sigma\in\widetilde{G}}\sigma(y_{i}v)\in V^{G}$, for $i=1,\dots,n$.

We have then obtained that the $\mathcal{C}_{K}$-vector space $V^{G}$ has
dimension $r=\dim_{\overline{\mathcal{C}_{K}}}V$. Now $V^{G}$ is clearly a
vector space of solutions to (2.1). Let us define
$L:=K\big{\langle}V^{G}\big{\rangle}$ the subfield of $\mathcal{L}$ generated
by the elements in $V^{G}$. By construction, $L$ is a Picard–Vessiot extension
of $K$ for (2.1). Since $L\subset L_{D}$, the field $L$ is a formally real
(resp. formally $p$-adic) field.

The complete diagram of field extensions is as follows:

$\displaystyle\begin{split}&\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
7.60416pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern 31.88194pt\raise
0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{\mathcal{L}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
75.40984pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise 0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern 112.40994pt\raise
0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{\mathcal{L}_{D}\simeq\overline{L}_{D}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{}{\hbox{\lx@xy@droprule}}{\hbox{\lx<EMAIL_ADDRESS>0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{\overline{K}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
35.00694pt\raise-38.16887pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern
70.51389pt\raise-38.16887pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{\overline{K}_{D}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{}{\hbox{\lx@xy@droprule}}{\hbox{\lx<EMAIL_ADDRESS>0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern
31.60416pt\raise-76.33774pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{L\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
75.40984pt\raise-76.33774pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern
120.91702pt\raise-76.33774pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{L_{D}}$}}}}}}}{\hbox{\kern-7.60416pt\raise-115.08438pt\hbox{\hbox{\kern
0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{K\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{\hbox{\lx@xy@drawline@}}{\hbox{\kern
35.00694pt\raise-115.08438pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern
68.40973pt\raise-115.08438pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{K_{D}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces}}}}\ignorespaces\end{split}$
∎

######  Theorem 3.3.

Let $L_{1}$, $L_{2}$ denote two formally real Picard–Vessiot extensions of $K$
for (2.1). Suppose that $L_{1}$ and $L_{2}$ have orderings which induce the
same ordering on $K$. Then $L_{1}$ and $L_{2}$ are $K$-differentially
isomorphic.

###### Proof.

We have that $(L_{1})_{D}$ and $(L_{2})_{D}$ are two formally real
Picard–Vessiot extensions of $K_{D}$ for the system (3.1). We extend the
ordering from $K$ to $K_{D}$ and from $L_{i}$ to $(L_{i})_{D}$, $i=1,2$, in
such a way that the ordering of $(L_{i})_{D}$ restricts to the ordering in
$K_{D}$, $i=1,2$ (see Remark 3.1). By Theorem 2.19, we have a differential
$K_{D}$-isomorphism

$\displaystyle\varphi\colon\ L_{1}\langle u_{1},\dots,u_{m}\rangle\rightarrow
L_{2}\langle u_{1},\dots,u_{m}\rangle.$

If $L_{1}=K(y_{ij})$ and $L_{2}=K\big{(}\widetilde{y}_{ij}\big{)}$ for
fundamental matrices $(y_{ij})$ and $\big{(}\widetilde{y}_{ij}\big{)}$ of the
system (2.1), then $(\varphi(y_{ij}))_{1\leq i,j\leq r}$ is a fundamental
matrix of (2.1) which gives
$(\varphi(y_{ij}))=\big{(}\widetilde{y}_{ij}\big{)}C$, with $C\in M_{r\times
r}(\mathcal{C}_{K})$. Therefore $\varphi$ restricts to a differential
$K$-isomorphism

$\displaystyle\varphi_{|L_{1}}\colon\ L_{1}\rightarrow L_{2}.$ ∎

######  Theorem 3.4.

Let $L_{1}$, $L_{2}$ denote two formally $p$-adic Picard–Vessiot extensions of
$K$ for (2.1). Suppose that $L_{1}$ and $L_{2}$ have $K$-isomorphic $p$-adic
closures $L_{1}^{+}$ and $L_{2}^{+}$. Then $L_{1}$ and $L_{2}$ are
$K$-differentially isomorphic.

###### Proof.

As $u_{1},\dots,u_{m}$ are independent differential indeterminates, the
$K$-isomorphism from $L_{1}^{+}$ to $L_{2}^{+}$ may be extended to a field
isomorphism from $\big{(}L_{1}^{+}\big{)}_{D}$ to
$\big{(}L_{2}^{+}\big{)}_{D}$ over $K_{D}$. Now we may extend the $p$-adic
valuation to $\big{(}L_{1}^{+}\big{)}_{D}$ and define a $p$-valuation on
$\big{(}L_{2}^{+}\big{)}_{D}$ by transferring to it the valuation of
$\big{(}L_{1}^{+}\big{)}_{D}$ by means of the isomorphism. Next we choose a
$p$-adic closure $M_{1}$ of $\big{(}L_{1}^{+}\big{)}_{D}$. We can extend the
isomorphism from $\big{(}L_{1}^{+}\big{)}_{D}$ to
$\big{(}L_{2}^{+}\big{)}_{D}$ to an embedding of $M_{1}$ into an algebraic
closure of $\big{(}L_{2}^{+}\big{)}_{D}$. Its image, equipped with the
valuation induced by the valuation of $\big{(}L_{1}^{+}\big{)}_{D}$ via the
embedding is a $p$-adic closure $M_{2}$ of $\big{(}L_{2}^{+}\big{)}_{D}$. Now
$M_{1}$ and $M_{2}$ are isomorphic as valued fields over $K_{D}$ and are
$p$-adic closures of $(L_{1})_{D}$ and $(L_{2})_{D}$. By Theorem 2.19, we have
a differential $K_{D}$-isomorphism

$\displaystyle\varphi\colon\ L_{1}\langle u_{1},\dots,u_{m}\rangle\rightarrow
L_{2}\langle u_{1},\dots,u_{m}\rangle.$

The rest of the proof is like for Theorem 3.3. ∎

###  3.2 Galois correspondence

In the case when the field of constants of the differential field $K$ is
algebraically closed, the differential Galois group of a Picard–Vessiot
extension $L/K$ is defined as the group of $K$-differential automorphisms of
$L$. The next example illustrates that, when the field of constants is not
algebraically closed, the group of $K$-differential automorphisms of $L$ may
be “too small” to obtain a satisfactory differential Galois theory.

######  Example 3.5.

Let $K:=\mathbb{R}(t_{1},t_{2})$ be the field of rational functions in the
variables $t_{1}$, $t_{2}$ over the field $\mathbb{R}$ of real numbers endowed
with the usual derivations $\partial_{1}:=\partial/\partial t_{1}$ and
$\partial_{2}:=\partial/\partial t_{2}$. Clearly the field of constants of $K$
is $\mathbb{R}$. We consider the differential system

$\displaystyle\partial_{1}Y=t_{2}\,Y,\qquad\partial_{2}Y=t_{1}\,Y,$ (3.2)

defined over $K$. A solution to the system (3.2) is ${\rm e}^{t_{1}t_{2}}$,
hence $L:=K\big{(}{\rm e}^{t_{1}t_{2}}\big{)}$ is a formally real
Picard–Vessiot extension of $K$ for (3.2). If $\sigma$ is a $K$-differential
automorphism of $L$, we have $\sigma\big{(}{\rm
e}^{t_{1}t_{2}}\big{)}=\lambda{\rm e}^{t_{1}t_{2}}$, with
$\lambda\in\mathbb{R}\setminus\\{0\\}$ and composing two such automorphisms
amounts to multiplying the corresponding factors $\lambda$. Hence the group
$\operatorname{DAut}_{K}L$ of $K$-differential automorphisms of $L$ is
isomorphic to the multiplicative group of $\mathbb{R}$. We consider now the
intermediate field $F:=K\big{(}{\rm e}^{3t_{1}t_{2}}\big{)}$. An
$F$-differential automorphism $\tau$ of $L$ is given by $\tau\big{(}{\rm
e}^{t_{1}t_{2}}\big{)}=\lambda{\rm e}^{t_{1}t_{2}}$, with $\lambda^{3}=1$,
since $\tau$ must fix ${\rm e}^{3t_{1}t_{2}}$. Hence the group of
$F$-differential automorphisms of $L$ is trivial. The subgroups of
$\operatorname{DAut}_{K}L$ corresponding to $F$ and $L$ by the Galois
correspondence will then be equal whereas $F$ and $L$ are not equal.

Let $K$ be a formally real (resp. formally $p$-adic) partial differential
field with real closed (resp. $p$-adically closed) field of constants
$\mathcal{C}_{K}$ and let $L/K$ be a Picard–Vessiot extension. Let
$\overline{\mathcal{C}_{K}}$ denote an algebraic closure of $\mathcal{C}_{K}$
and consider the fields
$\overline{K}:=K\otimes_{\mathcal{C}_{K}}\overline{\mathcal{C}_{K}}$ and
$\overline{L}:=L\otimes_{\mathcal{C}_{K}}\overline{\mathcal{C}_{K}}$. As in
the ordinary case (see [8]), we shall consider the set
$\operatorname{DHom}_{K}\big{(}L,\overline{L}\big{)}$ of $K$-differential
morphisms from $L$ into $\overline{L}$ and transfer the group structure from
$\operatorname{DAut}_{\overline{K}}\overline{L}$ to
$\operatorname{DHom}_{K}\big{(}L,\overline{L}\big{)}$ by means of the
bijection

$\displaystyle\operatorname{DAut}_{\overline{K}}\overline{L}$ $\displaystyle\
\rightarrow\ \operatorname{DHom}_{K}\big{(}L,\overline{L}\big{)},$
$\displaystyle\tau$ $\displaystyle\ \mapsto\ \tau_{|L}.$

######  Definition 3.6.

For the Picard–Vessiot extension $L/K$ we define the differential Galois group
as the set $\operatorname{DHom}_{K}\big{(}L,\overline{L}\big{)}$ endowed with
the group structure given above and denote it by $\operatorname{DGal}(L|K)$.

######  Proposition 3.7.

The map $\operatorname{DGal}(L_{D}|K_{D})\rightarrow\operatorname{DGal}(L|K)$
which to each $K_{D}$-differential morphism from $L_{D}$ into
$\overline{L}_{D}=L_{D}\otimes_{\mathcal{C}_{K}}\overline{\mathcal{C}_{K}}$
assigns its restriction to $L$ is an isomorphism of groups.

###### Proof.

Since $L/K$ is a Picard–Vessiot extension, we have that
$L=K\big{(}\\{y_{ij}\\}_{1\leq i,j\leq r}\big{)}$, for $(y_{ij})_{1\leq
i,j\leq r}$ a fundamental matrix for some differential system $S$ defined over
$K$. If $\sigma$ is a $K_{D}$-differential morphism from $L_{D}$ into
$\overline{L}_{D}$, then $\sigma((y_{ij}))$ is a fundamental matrix for $S$.
Therefore there exists an invertible matrix $C$ with entries in the field of
constants $\overline{\mathcal{C}_{K}}$ of $\overline{L}_{D}$ such that
$\sigma((y_{ij}))=(y_{ij})C$, hence $\sigma((y_{ij}))$ has entries in
$\overline{L}$. By restricting $\sigma$ to $L$, we obtain then a $K$-morphism
from $L$ to $\overline{L}$. For an element $x$ in $L$, we have
$Dx=\sum_{i=1}^{m}u_{i}\partial_{i}x$ and
$\sigma(Dx)=\sum_{i=1}^{m}u_{i}\sigma(\partial_{i}x)$, since $u_{i}\in K_{D}$,
$1\leq i\leq m$. On the other hand
$D(\sigma(x))=\sum_{i=1}^{m}u_{i}\partial_{i}(\sigma(x))$. Since the elements
$u_{i}$ are algebraically independent, we obtain that $\sigma_{|L}$ commutes
with $\partial_{i}$, $1\leq i\leq m$. Hence $\sigma_{|L}$ is a
$K$-differential morphism from $L$ to $\overline{L}$. Reciprocally, a
$K$-differential morphism from $L$ to $\overline{L}$ may be extended to a
$K_{D}$-differential morphism from $L_{D}$ to $\overline{L_{D}}$. ∎

######  Theorem 3.8.

The differential Galois group $\operatorname{DGal}(L|K)$ is a
$\mathcal{C}_{K}$-defined closed subgroup of some general linear group defined
over $\overline{\mathcal{C}_{K}}$, i.e., a linear algebraic group defined over
$\mathcal{C}_{K}$. The $\mathcal{C}_{K}$-valued points of
$\operatorname{DGal}(L|K)$ correspond to the $K$-differential automorphisms of
$L$.

###### Proof.

The first assertion follows from Proposition 3.7 and [8, Proposition 1]. The
second one is clear from the definition of $\operatorname{DGal}(L|K)$. ∎

For a closed subgroup $H$ of $\operatorname{DGal}(L|K)$, $L^{H}$ is a partial
differential subfield of $L$ containing $K$. If $E$ is an intermediate partial
differential field, i.e., $K\subset E\subset L$, then $L/E$ is a
Picard–Vessiot extension and $\operatorname{DGal}(L|E)$ is a
$\mathcal{C}_{K}$-defined closed subgroup of $\operatorname{DGal}(L|K)$. We
obtain a Galois correspondence theorem.

######  Theorem 3.9.

Let $K$ be a formally real $($resp. formally $p$-adic$)$ partial differential
field with real closed $($resp. $p$-adically closed$)$ field of constants
$\mathcal{C}_{K}$, let $L/K$ be a Picard–Vessiot extension and
$\operatorname{DGal}(L|K)$ be its differential Galois group.

  1. $1.$

The correspondences

$\displaystyle H\mapsto L^{H},\qquad E\mapsto\operatorname{DGal}(L|E)$

define inclusion inverting mutually inverse bijective maps between the set of
$\mathcal{C}_{K}$-defined closed subgroups $H$ of $\operatorname{DGal}(L|K)$
and the set of partial differential fields $E$ with
$K\\!\subset\\!E\\!\subset\\!L$.

  2. $2.$

The intermediate partial differential field $E$ is a Picard–Vessiot extension
of $K$ if and only if the subgroup $\operatorname{DGal}(L|E)$ is normal in
$\operatorname{DGal}(L|K)$. In this case, the restriction morphism

$\displaystyle\operatorname{DGal}(L|K)$ $\displaystyle\ \rightarrow\
\operatorname{DGal}(E|K),$ $\displaystyle\sigma$ $\displaystyle\ \mapsto\
\sigma_{|E}$

induces an isomorphism

$\displaystyle\operatorname{DGal}(L|K)/\operatorname{DGal}(L|E)\simeq\operatorname{DGal}(E|K).$

###### Proof.

The proof follows the same steps as the one of [8, Theorem 1]. In the present
case, we use the Galois correspondence theorem for partial differential fields
with an algebraically closed field of constants (see [24, Appendix D]). ∎

###  3.3 Liouvillian extensions

In this section we restrict to formally real partial differential fields. We
note that real Liouville functions may be characterised by topological
properties (see [11]) and it is therefore interesting to characterise real
Liouville solutions to differential systems defined over formally real fields
by means of the differential Galois group.

######  Definition 3.10.

Let $K$ be a field endowed with pairwise commuting derivations
$\partial_{1},\dots,\partial_{m}$. Let $L/K$ be a partial differential field
extension, $\alpha$ an element in $L$. We say that $\alpha$ is

  1. $-$

_an integral over $K$_ if $\partial_{k}\alpha=a_{k}\in K$, $1\leq k\leq m$,
and $a_{k}$ is not a derivative in $K$ for all $k$;

  2. $-$

_the exponential of an integral over $K$_ if $\partial_{k}\alpha/\alpha\in
K\setminus\\{0\\}$, $1\leq k\leq m$.

######  Remark 3.11.

If $\partial_{k}\alpha=a_{k}\in K$, for $1\leq k\leq m$, then
$D\alpha=\sum_{k=1}^{m}u_{k}a_{k}\in K_{D}$ and if $a_{k}$ is not a derivative
in $K$ for all $k$, then $\sum_{k=1}^{m}u_{k}a_{k}$ is not a derivative in
$K_{D}$. So, if $\alpha$ is an integral over $K$, it is also an integral over
$K_{D}$. Analogously, if $\partial_{k}\alpha/\alpha\in K\setminus\\{0\\}$, for
$1\leq k\leq m$, then $D\alpha/\alpha\in K_{D}\setminus\\{0\\}$. So, if
$\alpha$ is the exponential of an integral over $K$, it is also the
exponential of an integral over $K_{D}$.

Let now $K$ be a formally real field with real closed field of constants
$\mathcal{C}_{K}$. From Proposition 3.7 and the corresponding results in the
ordinary case [7, Examples 7 and 8], we obtain that, if $\alpha$ is an
integral over $K$, then $K\langle\alpha\rangle/K$ is a real Picard–Vessiot
extension and its differential Galois group
$\operatorname{DGal}(K\langle\alpha\rangle|K)$ is isomorphic to the additive
group $\mathbb{G}_{a}$; and, if $\alpha$ is the exponential of an integral and
the field $K\langle\alpha\rangle$ is real and with field of constants equal to
$\mathcal{C}_{K}$, then $K\langle\alpha\rangle/K$ is a Picard–Vessiot
extension and $\operatorname{DGal}(K\langle\alpha\rangle|K)$ is isomorphic to
the multiplicative group $\mathbb{G}_{m}$, or a finite subgroup of it.

######  Definition 3.12.

A partial differential field extension $L/K$ is called a _Liouvillian
extension_ (resp. a _generalised Liouvillian extension_) if there exists a
chain of intermediate partial differential fields $K=F_{1}\subset
F_{2}\subset\dots\subset F_{n}=L$ such that $F_{i+1}=F_{i}(\alpha_{i})$, where
$\alpha_{i}$ is either an integral or the exponential of an integral over
$F_{i}$ (resp. or $\alpha_{i}$ is algebraic over $F_{i}$), $1\leq i\leq n-1$.

######  Lemma 3.13.

If $L/K$ is a $($generalised$)$ Liouvillian extension of partial differential
fields then the ordinary differential field extension $L_{D}/K_{D}$ is a
$($generalised$)$ Liouvillian extension.

###### Proof.

Let $K=F_{1}\subset F_{2}\subset\dots\subset F_{n}=L$ be a chain of
intermediate partial differential fields such that
$F_{i+1}=F_{i}(\alpha_{i})$. We consider the chain of ordinary differential
fields $K_{D}\subset(F_{2})_{D}\subset\dots\subset L_{D}$. We have
$(F_{i+1})_{D}=F_{i+1}\langle
u_{1},\dots,u_{m}\rangle=F_{i}(\alpha_{i})\langle
u_{1},\dots,u_{m}\rangle=F_{i}\langle
u_{1},\dots,u_{m}\rangle(\alpha_{i})=(F_{i})_{D}(\alpha_{i})$. Now, by Remark
3.11, if $\alpha_{i}$ is an integral or the exponential of an integral or
algebraic over $F_{i}$, then $\alpha_{i}$ is an integral or the exponential of
an integral or algebraic over $(F_{i})_{D}$, respectively. ∎

######  Definition 3.14.

Let $G$ be a connected solvable linear algebraic group defined over a field
$\mathcal{C}$. We say that $G$ is $\mathcal{C}$-split if it has a composition
series $G=G_{1}\supset G_{2}\supset\cdots\supset G_{s}=1$ consisting of
connected $\mathcal{C}$-defined closed subgroups such that $G_{i}/G_{i+1}$ is
$\mathcal{C}$-isomorphic to $\mathbb{G}_{a}$ or $\mathbb{G}_{m}$, $1\leq i<s$.

From the results obtained in the ordinary case [7, Section 3, Theorems 17 and
18], we obtain the characterisation of real Liouvillian extensions of real
partial differential fields.

######  Theorem 3.15.

Let $K$ be a real partial differential field with real closed field of
constants $\mathcal{C}_{K}$, $L/K$ be a formally real Picard–Vessiot
extension, $\operatorname{DGal}(L|K)$ be its differential Galois group. The
following conditions are equivalent:

  1. $1.$

$L/K$ is a generalised Liouvillian extension.

  2. $2.$

$L$ is contained in a generalised Liouvillian extension $M$ of $K$, such that
the field of constants of $M$ is $\mathcal{C}_{K}$ and $M$ is a formally real
field.

  3. $3.$

The identity component of $\operatorname{DGal}(L|K)$ is solvable and
$\mathcal{C}_{K}$-split.

###### Proof.

Clearly (1) implies (2). By Lemma 3.13, (2) implies that $L_{D}$ is contained
in a formally real generalised Liouvillian extension of $K_{D}$, not adding
constants to $K_{D}$. Then by Proposition 3.7 and [7, Theorem 18], we obtain
(3).

We assume now that the identity component $G^{0}$ of
$G:=\operatorname{DGal}(L|K)$ is solvable and $\mathcal{C}_{K}$-split and let
$L^{0}=L^{G^{0}}$. Since $\big{[}G:G^{0}\big{]}$ is finite, $L^{0}/K$ is a
finite extension. By [2, Chapter V, Theorem 15.4], in $G^{0}$ is
triangularizable by a matrix with entries in $\mathcal{C}_{K}$. We may then
assume that there exist elements $v_{1},\dots,v_{r}\in L$ such that
$L=L^{0}\langle v_{1},\dots,v_{r}\rangle$ and for every $\sigma\in G^{0}$ we
have

$\displaystyle\sigma
v_{j}=a_{1j}v_{1}+\dots+a_{j-1,j}v_{j-1}+a_{jj}v_{j},\qquad j=1,\dots,r,$
(3.3)

with $a_{ij}$ constants in $L(i)$ (depending on $\sigma$). The first equality
is $\sigma v_{1}=a_{11}v_{1}$ which implies
$\sigma(\partial_{k}v_{1}/v_{1})=\partial_{k}v_{1}/v_{1}$, for all $\sigma\in
G^{0}$ and all $k=1,\dots,m$, hence $v_{1}$ is the exponential of an integral
over $L^{0}$. Now dividing the equations in (3.3) for $j=2$ to $r$ by the
first equation and applying $\partial_{k}$, we obtain

$\displaystyle\sigma\partial_{k}\bigg{(}\dfrac{v_{j}}{v_{1}}\bigg{)}=\dfrac{a_{2j}}{a_{11}}\partial_{k}\bigg{(}\dfrac{v_{2}}{v_{1}}\bigg{)}+\dots+\dfrac{a_{j-1,j}}{a_{11}}\partial_{k}\bigg{(}\dfrac{v_{j-1}}{v_{1}}\bigg{)}+\dfrac{a_{jj}}{a_{11}}\partial_{k}\bigg{(}\dfrac{v_{j}}{v_{1}}\bigg{)},\qquad
j=2,\dots,r,$

for $k=1,\dots,m$. By induction hypothesis on $r$, we obtain that the
extension

$\displaystyle L^{1}:=L^{0}\langle
v_{1}\rangle\bigg{\langle}\bigg{\\{}\partial_{k}\bigg{(}\dfrac{v_{j}}{v_{1}}\bigg{)}\bigg{\\}}_{k=1,\dots,m;j=2,\dots,r}\bigg{\rangle}\Big{/}L^{0}$

is a Liouvillian extension. Now the elements $v_{j}/v_{1}$ are integral over
$L^{1}$ and we obtain that $L/L^{0}$ is a Liouvillian extension and $L/K$ is a
generalised Liouvillian extension. ∎

######  Remark 3.16.

From Lemma 3.13, Proposition 3.7 and Theorem 3.15, we obtain that $L/K$ is a
formally real (generalised) Liouvillian extension of partial differential
fields if and only if the ordinary differential field extension $L_{D}/K_{D}$
is a formally real (generalised) Liouvillian extension.

######  Example 3.17.

Let $K:=\mathbb{R}(t_{1},t_{2})$ be the field of rational functions in the
variables $t_{1}$, $t_{2}$ over the field $\mathbb{R}$ of real numbers endowed
with the usual derivations $\partial_{1}:=\partial/\partial t_{1}$ and
$\partial_{2}:=\partial/\partial t_{2}$. The field of constants of $K$ is
$\mathbb{R}$. We consider the differential system

$\displaystyle\partial_{1}Y=\begin{pmatrix}1/t_{1}&0\\\
0&1/t_{1}\end{pmatrix}Y,\qquad\partial_{2}Y=\begin{pmatrix}0&1\\\
-1&0\end{pmatrix}Y,$ (3.4)

defined over $K$. A fundamental matrix for the system (3.4) is

$\displaystyle\begin{pmatrix}t_{1}\sin t_{2}&-t_{1}\cos t_{2}\\\ t_{1}\cos
t_{2}&t_{1}\sin t_{2}\end{pmatrix}\\!,$

hence $L:=K\langle t_{1}\sin t_{2},t_{1}\cos t_{2}\rangle=K\langle\sin
t_{2},\cos t_{2}\rangle$ is a Picard–Vessiot extension of $K$ for (3.4). An
element $\sigma$ in the Galois group $\operatorname{DGal}(L/K)$ is determined
by $\sigma(\sin t_{2})$ and $\sigma(\cos t_{2})$. If $\sigma(\sin
t_{2})\allowbreak=a\sin t_{2}+b\cos t_{2}$, then $\sigma(\cos
t_{2})=\sigma(\partial_{2}(\sin t_{2}))=\partial_{2}(\sigma(\sin t_{2}))=a\cos
t_{2}-b\sin t_{2}$. Moreover $(\sin t_{2})^{2}+(\cos t_{2})^{2}=1$ implies
$(\sigma(\sin t_{2}))^{2}+(\sigma(\cos t_{2}))^{2}=1$, which gives
$a^{2}+b^{2}=1$. We obtain then
$\operatorname{DGal}(L/K)\simeq\operatorname{SO}_{2}$.

We consider now the differential system

$\displaystyle\partial_{1}Y=\begin{pmatrix}1/t_{1}&0\\\
0&1/t_{1}\end{pmatrix}Y,\qquad\partial_{2}Y=\begin{pmatrix}0&1\\\
1&0\end{pmatrix}Y,$ (3.5)

defined over $K$. A fundamental matrix for the system (3.5) is

$\displaystyle\begin{pmatrix}t_{1}\sinh t_{2}&t_{1}\cosh t_{2}\\\ t_{1}\cosh
t_{2}&t_{1}\sinh t_{2}\end{pmatrix}\\!,$

hence $L:=K\langle t_{1}\sinh t_{2},t_{1}\cosh t_{2}\rangle=K\langle\sinh
t_{2},\cosh t_{2}\rangle=K\big{\langle}{\rm e}^{t_{2}}\big{\rangle}$ is a
Picard–Vessiot extension of $K$ for (3.5). An element $\sigma$ in the Galois
group $\operatorname{DGal}(L/K)$ is determined by $\sigma\big{(}{\rm
e}^{t_{2}}\big{)}$ and $\sigma\big{(}{\rm e}^{t_{2}}\big{)}=c{\rm e}^{t_{2}}$,
with $c\in\mathbb{R}\setminus\\{0\\}$. We obtain then
$\operatorname{DGal}(L/K)\simeq\mathbb{G}_{m}$.

We note that $\operatorname{SO}_{2}$ is isomorphic to $\mathbb{G}_{m}$ over
$\mathbb{C}$ but not over $\mathbb{R}$. Considering the systems (3.4) and
(3.5) as defined over the field $\mathbb{R}(t_{1},t_{2})$ we obtain that the
Picard–Vessiot extension for (3.5) is a real Liouvillian extension whereas the
Picard–Vessiot extension for (3.4) is not a real generalised Liouvillian
extension, since $\operatorname{SO}_{2}$ is not $\mathbb{R}$-split.111We note
that Remark 3 in [11] is incorrectly translated from the Russian original. The
third sentence must read: “This circle does not have a normal tower of
subgroups with quotient groups isomorphic either to the additive or the
multiplicative group of the field $\mathbb{R}$”.

##  4 Comments on the integrability of real dynamical systems

In this section we present the relationship between the concept of real
Liouville function and the results on Liouvillian extensions of formally real
differential fields treated in [7] and Section 3.3 of this paper. The
solutions of differential equations which until now were abstract elements are
considered in this section as real functions in real variables and we study
their properties as such functions. Considering them as elements of a field of
real functions, which is a formally real differential field, the results of
the preceding sections apply. We comment as well on some questions related
with the integrability of real dynamical systems going beyond the
Liouviallinity questions. It is worth noting that real Liouville functions are
a particular case of Pfaff functions which A. Khovanskii studies in his
monograph [16]. However Pfaff functions may appear as solutions to non-linear
differential equations and at present a Galois theory for non-linear
differential equations defined over a formally real differential field has not
been established. Khovanskii’s theory uses essentially the completeness of the
field $\mathbb{R}$ of real numbers. It would be interesting to give an
algebraic approach to this theory allowing to consider also real closed fields
different from the field of real numbers.

On the other hand, Theorem 3.15 shows that real Liouvillianity of an extension
$L/K$ of formally real differential fields is codified in the structure of the
differential Galois group $\operatorname{DGal}(L/K)$. A condition of this type
is suggested by A. Grothendieck in his proposal of an axiomatic definition of
tame topology (cf. [13, p. 272]). In our approach we use differential Galois
groups defined over the real closed field of constants.

###  4.1 Real Liouvillianity of solutions

To any open and connected subset $U\subset\mathbb{R}^{n}$ we assign the
integral domain of real analytic functions $\mathcal{O}(U)$ equipped with
standard partial derivations $\partial_{1}=\frac{\partial}{\partial
x_{1}},\dots,\partial_{n}=\frac{\partial}{\partial x_{n}}$. Let
$\mathcal{M}(U)$ denote the fraction field
$\operatorname{Fr}(\mathcal{O}(U))$. We identify the elements of
$\mathcal{M}(U)$ with the complete meromorphic fractions, i.e., the functions

$\displaystyle\phi\colon\ U\setminus\\{g=0\\}\ni
x\mapsto\frac{f(x)}{g(x)}\in\mathbb{R},$

which are maximal elements with respect to inclusion. From now on, we consider
only differential fields which are subfields of differential fields of the
form $(\mathcal{M}(U),\partial_{1},\dots,\partial_{n})$. Besides, we identify
the field $\mathbb{R}$ with the trivial differential structure with the
subfield of constant functions in $\mathcal{M}(U)$.

######  Definition 4.1.

Let $f\colon U\to\mathbb{R}$ be a real analytic function defined in an open
and connected subset $U\subset\mathbb{R}^{n}$. $f$ is called a _real Liouville
function_ (resp.  _generalised real Liouville function_) if it lies in some
formally real Liouvillian extension (resp. generalised formally real
Liouvillian extension) $K\subset\mathcal{M}(U)$ of the field
$\mathbb{R}\subset\mathcal{M}(U)$.

######  Example 4.2.

Let us consider the following dynamical system in $\mathbb{R}^{2}$.

$\displaystyle Y^{\prime}=\begin{pmatrix}0&1\\\ 1/x^{2}&0\end{pmatrix}Y,$

which corresponds to the second order linear differential equation
$y^{\prime\prime}-\big{(}1/x^{2}\big{)}y=0,$ defined over the differential
field $K:=\mathbb{R}(x)$, with the usual derivation ${\rm d}/{\rm d}x$.
Applying Kovacic’s algorithm (see [5, Section 7.3]), we obtain that the two
following real functions form a basis of the vector space of solutions

$\displaystyle y_{1}=x^{(1+\sqrt{5})/2},\qquad y_{2}=x^{(1-\sqrt{5})/2}.$

A Picard–Vessiot extension for this system over $K$ is then $K\langle
y_{1},y_{2}\rangle=K\langle y_{1}\rangle$, since $y_{1}y_{2}=x\\!\in\\!K$. A
$K$-differential isomorphism from $K\langle y_{1}\rangle$ to
$\mathbb{C}(x)\langle y_{1}\rangle$ sends $y_{1}$ to $\lambda y_{1}$ with
$\lambda\in\mathbb{C}^{\ast}$. Hence the differential Galois group is the
multiplicative group $\mathbb{G}_{m}$, defined over $\mathbb{R}$.

Let us observe that the real Liouville functions $y_{1}$ and $y_{2}$ fulfill
Pfaff equations of the form $Y^{\prime}=\alpha\frac{Y}{x}$, where
$\alpha=\frac{1}{2}\big{(}1\pm\sqrt{5}\big{)}$ and therefore they can be
studied by the methods of Pfaff geometry. More precisely the graphs of $y_{1}$
and $y_{2}$ are leaves of the foliations in $\mathbb{R}^{2}$ defined by the
1-forms $\omega_{\alpha}=\alpha y{\rm d}x-x{\rm d}y$. Considering them in the
positive half plane $\\{x>0\\}$ we obtain separating solutions to which
Khovanskii’s theory is aplicable (cf. [16, pp. 4 and 5]). In this paper we do
not enter in details of this theory but an interested reader can consult [16].

Our next example is related with real Liouville first integrals of simple
gradient systems. Recently gradient systems are intensively studied in
relation with the gradient conjecture and its generalisations (see [3, 4]).

######  Example 4.3.

T.H. Colding and W.P. Minicozzi II in their recent work [3] have formulated a
generalised version of the famous René Thom conjecture. More precisely
Conjecture 1.1 from [3] has been proved in [19], while the following
conjecture [3, Conjecture 1.2], called by the authors Arnold–Thom conjecture,
remains open.

Let $f$ be a real analytic function in an open set $U\subset\mathbb{R}^{n}$
and $\operatorname{grad}f$ be its gradient in the Euclidean metric.

######  Arnold–Thom conjecture.

If a gradient flow line $x(t)$ has a limit point $x_{0}\in U$, then the limit
of the unit tangents $\frac{x^{\prime}(t)}{|x(t)|}$ at $x_{0}$ exists.

In our second example, we present a polynomial gradient system that admits
Liouville first integral. Its trajectories in the neighborhood of the origin
are $C^{1}$ manifolds with boundary, but do not admit a $C^{2}$ extension
through their limit point. Thus this example shows that the above statement of
Arnold–Thom conjecture is optimal.

Let us consider a simple quadratic polynomial potential on $\mathbb{R}^{2}$

$\displaystyle f(x,y)=\lambda x^{2}+\mu
y^{2},\qquad\lambda,\mu\in\mathbb{R}\setminus\\{0\\}$

and its gradient dynamical system

$\displaystyle\frac{{\rm d}x}{{\rm d}t}=2\lambda x,$ $\displaystyle\frac{{\rm
d}y}{{\rm d}t}=2\mu y.$ (4.1)

Solving (4.1), one obtains the following real Liouvillian first integral

$I(x,y)=\dfrac{x^{\mu}}{y^{\lambda}}.$

It is easy to see that for $\lambda=2$, $\mu=3$ and $I(x,y)=1$, the level
curve is an ordinary cusp. Moreover as the curvature at the origin of
$\big{\\{}(x,y)\in\mathbb{R}^{2}\colon y^{2}=x^{3}\big{\\}}$ tends to
infinity, a $C^{2}$ prolongation of the branch curve $y=x^{3/2}$ is impossible
(see, e.g., [10, Theorem 5.1.6]).

###  4.2 Final remarks

The Arnold–Thom conjecture was known at the beginning of the 90’s of last
century under the name “Tangent Problem” (see [20, Section 9]). One more
interesting conclusion suggested by the Example 4.3 is that rational or, more
generally, meromorphic integrability is quite exceptional even among systems
that admit Liouville integrals. An interested reader can consult the excellent
survey on the classical integrability problems [21] and some more
computational examples of non-integrable real dynamical systems in [14].

## Acknowledgments

We are very thankful to the anonymous referees for their valuable comments
which helped us to improve significantly the presentation of our results. R.
Mohseni acknowledges support of the Polish Ministry of Science and Higher
Education. T. Crespo and Z. Hajto acknowledge support of grant
PID2019-107297GB-I00 (MICINN).

## References

  * [1] Bochnak J., Coste M., Roy M.-F., Real algebraic geometry, Ergebnisse der Mathematik und ihrer Grenzgebiete (3), Vol. 36, Springer-Verlag, Berlin, 1998\.
  * [2] Borel A., Linear algebraic groups, 2nd ed., Graduate Texts in Mathematics, Vol. 126, Springer-Verlag, New York, 1991.
  * [3] Colding T.H., Minicozzi II W.P., Arnold–Thom gradient conjecture for the arrival time, Comm. Pure Appl. Math. 72 (2019), 1548–1577, arXiv:1712.05381.
  * [4] Colding T.H., Minicozzi II W.P., Analytical properties for degenerate equations, in Geometric analysis, Progr. Math., Vol. 333, Birkhäuser/Springer, Cham, 2020, 57–70, arXiv:1804.08999.
  * [5] Crespo T., Hajto Z., Algebraic groups and differential Galois theory, Graduate Studies in Mathematics, Vol. 122, Amer. Math. Soc., Providence, RI, 2011.
  * [6] Crespo T., Hajto Z., Picard–Vessiot theory and the Jacobian problem, Israel J. Math. 186 (2011), 401–406.
  * [7] Crespo T., Hajto Z., Real Liouville extensions, Comm. Algebra 43 (2015), 2089–2093, arXiv:1206.2283.
  * [8] Crespo T., Hajto Z., Sowa-Adamus E., Galois correspondence theorem for Picard–Vessiot extensions, Arnold Math. J. 2 (2016), 21–27, arXiv:1502.08026.
  * [9] Crespo T., Hajto Z., van der Put M., Real and $p$-adic Picard–Vessiot fields, Math. Ann. 365 (2016), 93–103, arXiv:1307.2388.
  * [10] Dubrovin B.A., Fomenko A.T., Novikov S.P., Modern geometry – methods and applications. Part I. The geometry of surfaces, transformation groups, 2nd ed., and fields, Graduate Texts in Mathematics, Vol. 93, Springer-Verlag, New York, 1992.
  * [11] Gel’fond O.A., Khovanskii A.G., Real Liouville functions, Funct. Anal. Appl. 14 (1980), 122–123.
  * [12] Gillet H., Gorchinskiy S., Ovchinnikov A., Parameterized Picard–Vessiot extensions and Atiyah extensions, Adv. Math. 238 (2013), 322–411, arXiv:1110.3526.
  * [13] Grothendieck A., Esquisse d’un programme, in Geometric Galois Actions, 1, Editors L. Schneps, P. Lochak, London Math. Soc. Lecture Note Ser., Vol. 242, Cambridge University Press, Cambridge, 1997, 5–48, English translation on pp. 243–283.
  * [14] Hajto Z., Mohseni R., Tame topology and non-integrability of dynamical systems, arXiv:2008.12074.
  * [15] Kamensky M., Pillay A., Interpretations and differential Galois extensions, Int. Math. Res. Not. 2016 (2016), 7390–7413.
  * [16] Khovanskii A.G., Fewnomials, Translations of Mathematical Monographs, Vol. 88, Amer. Math. Soc., Providence, RI, 1991.
  * [17] Kolchin E.R., Algebraic matric groups and the Picard–Vessiot theory of homogeneous linear ordinary differential equations, Ann. of Math. 49 (1948), 1–42.
  * [18] Kolchin E.R., Picard–Vessiot theory of partial differential fields, Proc. Amer. Math. Soc. 3 (1952), 596–603.
  * [19] Kurdyka K., Mostowski T., Parusiński A., Proof of the gradient conjecture of R. Thom, Ann. of Math. 152 (2000), 763–792, arXiv:math.AG/9906212.
  * [20] Łojasiewicz S., On semi-analytic and subanalytic geometry, in Panoramas of Mathematics (Warsaw, 1992/1994), Banach Center Publ., Vol. 34, Polish Acad. Sci. Inst. Math., Warsaw, 1995, 89–104.
  * [21] Maciejewski A.J., Przybylska M., Differential Galois theory and integrability, Int. J. Geom. Methods Mod. Phys. 6 (2009), 1357–1390, arXiv:0912.1046.
  * [22] Prestel A., Lectures on formally real fields, Lecture Notes in Math., Vol. 1093, Springer-Verlag, Berlin, 1984.
  * [23] Prestel A., Roquette P., Formally $p$-adic fields, Lecture Notes in Math., Vol. 1050, Springer-Verlag, Berlin, 1984.
  * [24] van der Put M., Singer M.F., Galois theory of linear differential equations, Grundlehren der mathematischen Wissenschaften, Vol. 328, Springer-Verlag, Berlin, 2003.