Output dead beat control for a class of nonlinear discrete time systems, which are described by a single input-output (I-O) polynomial difference equation, is considered. The class of systems considered is restricted to systems with a two-dimensional state space description. It is assumed that the highest degree with which the present input appears in the equation is odd. Necessary and sufficient conditions for the existence of output dead beat control and for the stability of the zero output constrained dynamics are presented. We also design a minimum time output dead beat control algorithm (feedback controller) which yields stable zero dynamics, whenever this is feasible. A number of interesting phenomena are discussed and illustrated with examples.