Endomorphisms of Polynomial Rings and Jacobians

The Jacobian Conjecture is established : 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]. 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 conjecture can be stated as follows: 1991 Mathematics Subject Classification. Primary 13C20, Secondary 13F99.