A generalization of the Fujisawa–Kuh global inversion theorem

We discuss the problem of global invertibility of nonlinear maps defined on the finite dimensional Euclidean space via differential tests. We provide a generalization of the Fujisawa–Kuh global inversion theorem and introduce a generalized ratio condition which detects when the pre-image of a certain class of linear manifolds is non-empty and connected. In particular, we provide conditions that also detect global injectivity.