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In this paper, we consider local observability for polynomial systems. When testing for the local observability of nonlinear systems, the observability rank condition based on the inverse function theorem is commonly used. However, the observability rank condition is only a sufficient condition. Recently, a different viewpoint of the observability rank condition, a necessary and sufficient condition for local observability at an initial state, has been derived. However, it is still difficult to check the local observability condition for all initial states. In this paper, we derive a necessary condition for the local observability of polynomial systems that is based on the local observability condition at an initial state. The obtained condition is characterized by a finite set of equations because polynomial rings are Noetherian.