Effective difference elimination and Nullstellensatz
Alexey Ovchinnikov, Gleb Pogudin, Thomas Scanlon · Journal of the European Mathematical Society · 2020
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 \mathbf x = (x_1, \dots, x_m) and \mathbf u = u_1, \dots, u_r) , if these equations have any nontrivial consequences in the \mathbf 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.