nLab Picard-Vessiot theory

Idea

Picard-Vessiot theory is the Galois theory of homogeneous linear ordinary differential equations. A version for homogeneous linear difference equations has also been developed. Alternatively, terms differential Galois theory and difference Galois theory have also been used.

Literature

Classical references

• E. R. Kolchin, Algebraic matric groups and the Picard-Vessiot theory of homogeneous linear ordinary differential equations, Ann. Math. (2nd ser) 49:1 (1948) 1-42 jstor doi
• Charles H. Franke, Picard-Vessiot theory of linear homogeneous difference equations, Trans. Amer. Math. Soc. 108:3 (1963) 491-515 jstor doi

In categorical setup

Tannakian approach has been developed in

• Pierre Deligne, Catégories Tannakiennes, Grothendieck Festschrift, vol. II, Birkhäuser Progress in Math. 87 (1990) 111-195, pdf
• A. Ovchinnikov, Tannakian categories, linear differential algebraic groups, and parametrized linear differential equations, Transformation Groups 14, 195–223 (2009) doi

In the setup of differential abelian tensor categories over differential rings,

• Henri Gillet, Sergey Gorchinskiy, Alexey Ovchinnikov, Parameterized Picard–Vessiot extensions and Atiyah extensions, Adv. Math. 238 (2013) 322-411 doi

Galois theory for differential equations is one of the examples in

• Francis Borceux, George Janelidze, Galois Theories, Cambridge Studies in Advanced Mathematics 72, Cambridge University Press, 2001.

Difference equations are viewed from topos perspective (with sporadic recourse to Galois theory) in

• Ivan Tomašić, A topos-theoretic view of difference algebra, arxiv/2001.09075

Connections to and variants in model theory

• D. Marker, Model theory of differential fields, In Model theory of fields, edited by Marker, Messmer, Pillay. Lecture Notes in Logic 5. ASL-Peters, 2006
• Moshe Kamensky, Anand Pillay, Interpretations and differential Galois extensions, IMRN 24 (2016) 7390–7413 doi
• Z. Chatzidakis, Anand Pillay, Generalized Picard-Vessiot extensions and differential Galois cohomology, 2017, pdf

Hopf algebraic approach

• Mitsuhiro Takeuchi, A Hopf algebraic approach to the Picard-Vessiot theory, J. Algebra 122:2 (1989) 481-509 doi

…Hopf algebraic approach simplifies and generalizes the theory. Among other things the generalization gives a Picard-Vessiot theory for positive characteristic, for fields with not necessarily commuting derivations, and even for fields with (a set of) higher derivations.

• K. Amano, A. Masuoka, M. Takeuchi, Hopf algebraic approach to Picard-Vessiot theory, Handbook of Algebra, Volume 6, (2009) 127-171 doi