\(q\)-difference equations. (Équations aux \(q\)-différences.) (French) Zbl 1063.39015

Di Vizio, L.; Ramis, J.-P.; Sauloy, J.; Zhang, C.

Gaz. Math., Soc. Math. Fr. 96, 20-49 (2003).

The paper is, on the one hand, a short motivated historical introduction to \(q\)-difference equations; on the other hand, a survey of work done in this area, mostly during the last decades and mostly by the authors of the paper (including the present reviewer). The paper is written in French. The historical part mentions the origins of the subject (from Euler, Gauss, Heine, Jacobi, Ramanujan …) and the great breakthrough by G. D. Birkhoff at the beginning of the XXth century, tying it with what is nowadays called the Riemann-Hilbert correspondence.
The survey part covers mostly two areas. First, there is work by Ramis, Sauloy and Changgui Zhang over the complex field (local and global algebraic and analytical theory in the Fuchsian case; local algebraic and analytical theory in the irregular case); in this part, the main problems are resolution, classification and monodromy in an extended sense, without explicit Galois theory. Second, there is work done by Yves André and Lucia Di Vizio (and also Bezivin) over \(p\)-adic fields and algebraic number fields, dealing with the arithmetical sides of the theory (like the proof by Di Vizio of the \(q\)-analog of Grothendieck-Katz conjecture).

Reviewer: Jacques Sauloy (Toulouse)

MSC:

39A13	Difference equations, scaling (\(q\)-differences)
39-03	History of difference and functional equations
01A55	History of mathematics in the 19th century
01A60	History of mathematics in the 20th century

Keywords:

\(q\)-difference equations; Riemann-Hilbert correspondence; analytical theory; resolution; classification; monodromy; \(p\)-adic fields; algebraic number fields; Grothendieck-Katz conjecture; historical survey; research survey; local algebraic theory; global algebraic theory