A note on Meisters' and Olech's proof of the global asymptotic stability Jacobian conjecture
Let \(k\) be a field of characteristic zero, and let \(F: k^ n\to k^ n\) be a polynomial map. Assume that at every point \(x\in k^ n\), the Jacobian of \(F\) has rank \(n\). Then there exists a positive integer \(N\) such that, for every \(x\in k^ n\), the fiber \(F^{-1}(x)\) has at most \(N\) elements. This theorem was proved by \textit{G. Meisters} and \textit{C. Olech} [Analyse mathématique et applications, Contrib. Honneur J.-L. Lions, 373-381 (1988; Zbl 0668.34048)] for the case \(n=2\) and \(k=\mathbb{R}\) using topological methods and used for proving a result on global stability. The author gives a proof using the theory of \({\mathcal D}\)- modules of differential equations.
scientific article; zbMATH DE number 32709