Sauloy, Jacques Galois theory of \(q\)-difference equations: the “analytical” approach. (English) Zbl 1036.12007 Braaksma, B. L. J. (ed.) et al., Differential equations and the Stokes phenomenon. Proceedings of the conference, Groningen, Netherlands, May 28–30, 2001. Singapore: World Scientific, 277-292 (2002). Summary: We show how a function theoretic approach to the Galois theory of Fuchsian \(q\)-difference equations (in contrast to the purely algebraic approach due to M. van der Put and M. F. Singer [Galois theory of difference equations. Lecture Notes in Mathematics. 1666. Berlin: Springer (1997; Zbl 0930.12006)] is possible and desirable. Based on classical complex geometry (theta functions and elliptic curves), it allows for a topological description of the Riemann-Hilbert-Birkhoff correspondence.For the entire collection see [Zbl 1007.00033]. Cited in 4 Documents MSC: 39A13 Difference equations, scaling (\(q\)-differences) 34M55 PainlevĂ© and other special ordinary differential equations in the complex domain; classification, hierarchies 12H10 Difference algebra Keywords:Galois theory; Fuchsian \(q\)-differnce equations; theta functions; elliptic curves; Riemann-Hilbert-Birkhoff correspondence Citations:Zbl 0930.12006 PDFBibTeX XMLCite \textit{J. Sauloy}, in: Differential equations and the Stokes phenomenon. Proceedings of the conference, Groningen, Netherlands, May 28--30, 2001. Singapore: World Scientific. 277--292 (2002; Zbl 1036.12007)