Normal Forms in Differential Galois Theory for the Classical Groups

2022/04/13 by Daniel Robertz, Robertz, Daniel, Matthias Seiß +1
Mathematics · #Advanced Topics in Algebra #Homotopy and Cohomology in Algebraic Topology #Algebraic Geometry and Number Theory

paper · pdf · doi:10.48550/arxiv.2204.06494

Abstract

Let G be a classical group of dimension d and let \boldsymbola=(a₁,…,a_d) be differential indeterminates over a differential field F of characteristic zero with algebraically closed field of constants C. Further let A(\boldsymbola) be a generic element in the Lie algebra \mathfrakg(F⟨ \boldsymbola ⟩) of G obtained from parametrizing a basis of \mathfrakg with the indeterminates \boldsymbola. It is known (cf. work by Juan) that the differential Galois group of \boldsymboly'=A(\boldsymbola)\boldsymboly over F⟨ \boldsymbola ⟩ is G(C). In this paper we construct a differential field extension L of F⟨ \boldsymbola ⟩ such that the field of constants of L is C, the differential Galois group of \boldsymboly'=A(\boldsymbola)\boldsymboly over L is still the full group G(C) and A(\boldsymbola) is gauge equivalent over L to a matrix in normal form which we introduced in work by Seiss. We also consider specializations of the coefficients of A(\boldsymbola).

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