Let R denote the ring Q or Z. (Not the reals!) Then R^n denotes the cartesian product R x ... x R (n times). Suppose f : R^n —> R^n is a polynomial mapping of the form f(x_1, ..., x_n) = (P_1(x_1,...,x_n), ..., P_n(x_1,...,x_n)) where each P_j is a polynomial in the x_1, ..., x_n over the ring R. Question: --------- (*) If f is a bijection, does that imply that each P_j is linear ? --------- The polynomial bijections of R^n —> R^n (R = Q or Z) form a group. Just in case there are any non-linear ones: What is this group? E.g. for Z^2 or Q^2. It is known that if the ring is the complex field C, there do exist non-linear polynomial bijections C^2 —> C^2 of form f(z,w) = (P(z,w), Q(z,w)). Of course the same question (*) can be asked about any ring at all. —Dan