Abstract This paper presents an algebraic algorithm for the calculation of the zero dynamics for nonlinear analytic MIMO state space systems. The algorithm is independent of the number of inputs and outputs and admits nonlinear dependences of the control signals, state variables, and outputs. Due to the fact that Grobner bases are applied, a restriction is made to systems for which dependences of the inputs, state space variables, and outputs are given by polynomial equations. The presented algorithm shows some similarity with existing geometrical formulations (Nijmeijer and van der Schaft, 1990; Isidori, 1995) and represents an algebraic interpretation.