Effective difference elimination and Nullstellensatz

We prove effective Nullstellensatz and elimination theorems for difference equations in sequence rings. More precisely, we compute an explicit function of geometric quantities associated to a system of difference equations (and these geometric quantities may themselves be bounded by a function of the number of variables, the order of the equations, and the degrees of the equations) so that for any system of difference equations in variables x = (x₁,..., x_m) and u = (u₁,..., u_r), if these equations have any nontrivial consequences in the x variables, then such a consequence may be seen algebraically considering transforms up to the order of our bound. Specializing to the case of m = 0, we obtain an effective method to test whether a given system of difference equations is consistent.