Endomorphisms of Polynomial Rings ; the Jacobian Conjecture

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: The Jacobian Conjecture. Let k be a field of characteristic zero, let k[X1, . . . , Xn] be a polynomial ring over k, and let k[f1, . . . , fn] be a subring of k[X1, . . . , Xn] generated by f1, . . . , fn over k. If the Jacobian det(∂fi/∂Xj) is a nonzero constant then k[X1, . . . , Xn] = k[f1, . . . , fn]. The Jacobian conjecture has been settled affirmatively in several cases. For example, Case(1) k(X1, . . . , Xn) is a Galois extension of k(f1, . . . , fn) (cf. [4],[6] and [14]); Case(2) deg fi ≤ 2 for all i (cf. [12] and [13]); Case(3) k[X1, . . . , Xn] is integral over k[f1, . . . , fn]. (cf. [4]). A general reference for the Jacobian Conjecture is [4]. Our objective of this paper is to give an affirmative answer to this conjecture. Throughout this paper, all fields, rings and algebras are assumed to be commutative with unity. For a ring R, R denotes the set of units of R and K(R) the total quotient ring. Our general reference for unexplained technical terms is [10]. 1. Etale Morphisms k → k and the Jacobian Conjecture In this section, we devote ourselves to proving the Jacobian Conjecture. 1991 Mathematics Subject Classification. Primary 13C20, Secondary 13F99.