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In this contribution global observability of nonlinear systems is investigated. The main idea is to derive a criterion which allows to decide if two initial states of a system can be distinguished by the output of the system. This well known criterion is equivalent to an infinite set of nonlinear equations. If the system is globally observable this set of equations has a very special solution set which can be characterized algebraically. Based on results from commutative algebra some new algorithmic methods to describe the solution set are given. This allows a global observability analysis for polynomial systems and for some specific classes of nonlinear systems. The new method is applied to an example from the literature.