  We introduce six new algebraic invariants for rational difference equations.
We use these invariants to perform a reduction of order in each case. This
reduction of order allows us to find forbidden sets in each case. These six
cases include two linear fractional rational difference equations of order
greater than one. In all six cases, we give a closed form solution for all
initial conditions which are not in the forbidden set. In all six cases, the
initial conditions and parameters are assumed to be arbitrary complex numbers.

Dynamical Systems