On definition of difference field, associated to nonlinear control system: several options

A universal inversive difference field that always exists is constructed for a submersive discrete-time nonlinear control system. This field is unique up to permutation of the control variables. Using the unique field in proofs will simplify them significantly. The construction extends the one for state-invertible systems, being the subset of submersive systems. An algorithm is given for finding independent variables of this field. It is proven that the algorithm stops in at most n + 1 steps, where n is the state dimension of the control system.