On the Jacobian Conjecture (original) (raw)

Taiwanese Journal of Mathematics

Let k be an algebraically closed field, and let f: k"-+ k" be a polynomial map. Then f is given by coordinate functions fl, f,, where each fi is a polynomial in n variables X1, X,. Iffhas a polynomial inverse g (g 1, ,), then the determinant of the Jacobian matrix f/OXj is a non-zero constant. This follows from the chain rule: Since f 0 is the identity, we have X 0i, f,), so X t=l This says that the product tXs is the identity matrix. Thus, the Jacobian determinant off is a non-vanishing polynomial, hence a constant. The Jacobian conjecture states, conversely, that if the characteristic of k is zero, and if f= (f,..., f,) is a polynomial map such that the Jacobian determinant is a non-zero constant, then f has a polynomial inverse. The problem first appeared in the literature (to my knowledge) in 1939 in [11] for k C. Many erroneous proofs have emerged, several of which have been published, all for k C, n 2. The conjecture is trivially true for n 1. For n > 1, the question is open. There has been a vigorous attempt by S. Abhyankar and T.-T. Moh to solve the problem for n 2. In this case it is known that the Jacobian conjecture is equivalent to the assertion that whenever f (f, f2) satisfies the Jacobian hypothesis, the total degree off divides that off2, or vice versa. Abhyankar and Moh have obtained a number of partial results by looking at the intersection of the curves fl and f2 at infinite in p2. Moh has proved, in fact, that the conjecture is true provided the degrees of A and A do not exceed 1 [15].