Algebraic nonsolvability of the problem of existence of holomorphic first integrals (Q1972697)

This very interesting work can be considered as a certain continuation of the works of Mattei and Houssu on the first integrals and holonomy. The author proves that the existence of an analytic first integral of the system \(\dot x= f(x,y)\), \(\dot y= g(x,y)\) with \(f,g\) analytic is an algebraically insolvable one (in the sense of Definition 2.1). This fact is announced in Theorem 2.3 and proved by showing the non-semialgebraicity of a certain set. As the problem of existence of the first integral in \(\mathbb{R}^2\) is equivalent to the same problem in \(\mathbb{C}^2\), the author begins with the complexification and treats the question in the complex case, using desingularisation and this in a very clear, explicit and natural way. The conjecture is made that no finite-order jet is sufficient to claim that any extension of it has a holomorphic first integral. This conjecture is proved in the nilpotent case (in the sense of Cerreau and Mattei) in Chapter 3 ``Perturbation of the holonomy in the nilpotent case''.