Coordinates transformations in nonlinear time-delay systems

When considering nonlinear time delay systems, the definition of an invertible change of coordinates is no more granted. Starting from a set of n − 1 exact differentials independent over K(d] in the variables x(t),... , x(t−sD), with D a constant commensurate delay and x an n-dimensional variable, we give necessary and sufficient conditions for the existence of an exact one-form such that to get a basis for $span_K(d] {dx[0]}$.