It is generally a very difficult problem to determine whether a given system of PDEs is consistent or not. One strategy is to think algebraically; that is, think of the system as algebraic polynomials and add algebraic equations for prolongations of all orders. This new system has infinitely many variables, and so now one can ask: is it enough to solve a finite subsystem of the latter? In this talk I will present new effective bounds that apply to this question. This is joint work with R. Gustavson.