A solution to Kronecker's problem

Consider ℤ[x₁,..., x_n], the multivariate polynomial ring over integers involving n variables. For a fixed n, we show that the ideal membership problem as well as the associated representation problem for ℤ[x₁,..., x_n] are primitive recursive. The precise complexity bounds are easily expressible by functions in the Wainer hierarchy. Thus, we solve a fundamental algorithmic question in the theory of multivariate polynomials over the integers. As a direct consequence, we also obtain a solution to certain foundational problem intrinsic to Kronecker's programme for constructive mathematics and provide an effective version of Hilbert's basis theorem. Our original interest in this area was aroused by Edwards' historical account of the Kronecker's problem in the context of Kronecker's version of constructive mathematics.