About the inverse problem in differential Galois theory: The differential Abhyankar conjecture.(English)Zbl 0860.12003

Braaksma, B. L. J. (ed.) et al., The Stokes phenomenon and Hilbert’s 16th problem. Proceedings of the workshop, Groningen, The Netherlands, May 31-June 3, 1995. Singapore: World Scientific. 261-278 (1996).
The author gives a description of his solution of two inverse problems in differential Galois theory: the local inverse problem and the inverse problem above an affine algebraic curve. The solution of this last problem parallels strongly the solution of the Abhyankar conjecture about coverings of affine algebraic curves in characteristic $$p>0$$ [see M. Raynaud, Invent. Math., 116, 425-462 (1994; Zbl 0798.14013)]. The following theorem describes a solution of the first problem.
Theorem. Let $$G$$ be a complex linear algebraic group. We denote by $$G^0$$ its identity component and by $$R_u$$ its unipotent radical. Let $$L(G)$$ be the subgroup of $$G$$ generated by all its maximal tori. It is a closed Zariski connected invariant subgroup. The following three statements are equivalent:
(i) The group $$G$$ is the differential Galois group of a Picard-Vessiot extension of $$C\{x\} [x^{-1}]$$.
(ii) The following three conditions hold: (a) the finite group $$G/G^0$$ is cyclic, (b) the dimension of $$R_u/[R_u,G^0]$$ is at most 1 and (c) the group $$G/G^0$$ acts trivially on $$R_u/[R_u,G^0]$$.
(iii) The quotient $$S'(G)=G/L(G)$$ is topologically generated by only one element. It differs slightly from a result which was described in [C. Mitschi and M. F. Singer, J. Pure Appl. Algebra 110, No. 2, 185-194 (1996; Zbl 0854.12001)].
For the entire collection see [Zbl 0846.00026].

MSC:

 12H05 Differential algebra 12F12 Inverse Galois theory 34M50 Inverse problems (Riemann-Hilbert, inverse differential Galois, etc.) for ordinary differential equations in the complex domain

Citations:

Zbl 0798.14013; Zbl 0854.12001