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Iterative differential Galois theory in positive characteristic: a model theoretic approach. (English) Zbl 1225.03041

Summary: This paper introduces a natural extension of Kolchin’s differential Galois theory to positive characteristic iterative differential fields, generalizing to the nonlinear case the iterative Picard-Vessiot theory recently developed by Matzat and van der Put. We use the methods and framework provided by the model theory of iterative differential fields. We offer a definition of strongly normal extension of iterative differential fields, and then prove that these extensions have good Galois theory and that a \(G\)-primitive element theorem holds. In addition, making use of the basic theory of arc spaces of algebraic groups, we define iterative logarithmic equations, finally proving that our strongly normal extensions are Galois extensions for these equations.

MSC:

03C60 Model-theoretic algebra
12H05 Differential algebra
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