On the Darboux integrability of polynomial differential systems
A method to find explicit closed forms for a first integral of a planar polynomial differential system from its invariant algebraic curves was given by \textit{G. Darboux} [C. R. LXXXVI, 581--586 (1878; JFM 10.0214.03)]. From that moment on, many research articles were devoted to the study of the integrability problem by using the invariant algebraic curves and its generalizations. This paper is a summary of all the results about this topic. The paper deals with systems in \(\mathbb{R}^n\) or \(\mathbb{C}^n\) with \(n \geq 2\). A very complete list of references together with the corresponding related results is included. The paper is, nonetheless, self-contained in such a way that all the main results are stated and a complete overview of the subject is given. Each section contains a different issue to which the Darboux integrability plays a role: exponential factors and the relation with the multiplicity of an invariant algebraic hypersurface, the Darboux theory of integrability in \(\mathbb{C}^n\), to take into accont the multiplicity at infinity, cosntruction of first integrals, Liouvillian integrability and some applications.
scientific article; zbMATH DE number 6134569
scientific article