On Endomorphisms of Polynomial Rings with Invertible Jacobians

The Jacobian Conjecture can be generalized and is established : Let S be a polynomial ring over a field of characteristic zero in finitely may variables. Let T be an unramified, finitely generated extension of S with T = k. Then T = S. Let k be an algebraically closed field, let k be an affine space of dimension n over k and let f : k −→ k be a morphism of algebraic varieties. Then f is given by coordinate functions f1, . . . , fn, where fi ∈ k[X1, . . . , Xn] and k = Max(k[X1, . . . , Xn]). If f has an inverse morphism, then the Jacobian det(∂fi/∂Xj) is a nonzero constant. This follows from the easy chain rule. The Jacobian Conjecture asserts the converse. If k is of characteristic p > 0 and f(X) = X +X, then df/dX = f (X) = 1 but X can not be expressed as a polynomial in f. Thus we must assume the characteristic of k is zero. The Jacobian Conjecture is the following : If f1, · · · , fn be elements in a polynomial ring k[X1, · · · , Xn] over a field k of characteristic zero such that det(∂fi/∂Xj) is a nonzero constant, then k[f1, · · · , fn] = k[X1, · · · , Xn]. To prove the Jacobian Conjecture, we treat a more general case. More precisely, we show the following result: 2000 Mathematics Subject Classification : Primary 13C25, Secondary 15A18