Elimination of unknowns for systems of algebraic differential-difference equations

We establish effective elimination theorems for ordinary differential-difference equations. Specifically, we find a computable function B(r,s) of the natural number parameters r and s so that for any system of algebraic ordinary differential-difference equations in the variables \bm {x} = x_1, \ldots , x_q and \bm {y} = y_1, \ldots , y_r, each of which has order and degree in \bm {y} bounded by s over a differential-difference field, there is a nontrivial consequence of this system involving just the \bm {x} variables if and only if such a consequence may be constructed algebraically by applying no more than B(r,s) iterations of the basic difference and derivation operators to the equations in the system. We relate this finiteness theorem to the problem of finding solutions to such systems of differential-difference equations in rings of functions showing that a system of differential-difference equations over \mathbb{C} is algebraically consistent if and only if it has solutions in a certain ring of germs of meromorphic functions.