Sufficiency of a necessary condition for local observability of discrete-time polynomial systems

We consider local observability of discrete-time polynomial systems. When testing local observability of nonlinear systems, the observability rank condition is commonly used. However, the rank condition is only a sufficient condition in general. For this problem, recently, in terms of commutative algebra and algebraic geometry, a necessary condition for local observability of the polynomial systems has been derived. In this paper, we show the necessary condition is a sufficient condition for local observability for almost all initial states. Based on this result, we derive a necessary and sufficient condition for local observability.