nLab differential Galois theory

Idea

Differential Galois theory is an analogue of Galois theory where fields are generalized to differential fields, hence is a theory for differential equations rather than just algebraic equations.

References

• Pierre Deligne, Catégories Tannakiennes, Grothendieck Festschrift, vol. II, Birkhäuser Progress in Math. 87 (1990) pp.111-195.

• Andy Magid, Differential Galois theory, Notices of the AMS (1999) pdf

• Teresa Crespo, The origins of differential Galois theory (pdf slides)

• Andy R. Magid, Universal covers and category theory in polynomial and differential Galois theory, pdf

Textbook accounts include

• Marius van der Put, Michael Singer, Galois theory of linear differential equations, Springer, Berlin (2003)

• M. Singer, Direct and inverse problems in differential Galois theory, in H. Bass, et al. (eds.) Selected works of Ellis Kolchin with commentary , Amer. Math. Soc. 1999, 527–554.

Discussion in terms of D-geometry is in

• João Pedro P. dos Santos, The behaviour of the differential Galois group on the generic and special fibres: A Tannakian approach, J. reine angew. Math. 637 (2009), 63-98 (pdf)

• João Pedro P. dos Santos, Lifting D-modules from positive to zero characteristic (pdf)

based on

• B. H. Matzat, M. van der Put, Iterative differential equations and the Abhyankar conjecture, J. Reine Angew. Math. 557 (2003), pp. 1–52.

A connection to geometric invariant theory is proposed in

• Yves Andre, Solution algebras of differential equations and quasi-homogeneous varieties, arxiv/1107.1179

Model theoretic aspects are discussed in

• Moshe Kamensky, Anand Pillay, Interpretations and differential Galois extensions, arxiv/1409.5915
• Anand Pillay, Differential Galois theory I, Illinois J. Math. 42(4) 1998
• D. Bertrand, A. Pillay, Galois theory, functional Lindemann-Weierstrass, and Manin maps, Pacific J. Math. 281:1 (2016) 51–82 doi