Eulerian operators and the Jacobian Conjecture, III (original) (raw)

Eulerian operators and the Jacobian conjecture

Proceedings of the American Mathematical Society, 1993

In this paper we introduce a new class of polynomial maps, the so-called nice polynomial maps. Using Eulerian operators we show how for these polynomial maps the main results obtained by Bass (Differential structure of étale extensions of polynomial algebras, Proc. Workshop on Commutative Algebra, MSRI, 1987) can be proved in a very simple and elementary way. Furthermore we show that every polynomial map F F satisfying the Jacobian condition, det J F ∈ k ∗ JF \in {k^{\ast }} , is equivalent to a nice polynomial map; more precisely the polynomial map F ( λ ) ( X ) = F ( X + λ ) − F ( λ ) {F_{(\lambda )}}(X) = F(X + \lambda ) - F(\lambda ) is nice for almost all λ ∈ k n \lambda \in {k^n} .

The Jacobian conjecture: ideal membership questions and recent advances

Affine Algebraic Geometry, 2005

CONTENTS 1. The Jacobian Conjecture 2. Ideals Defining the Jacobian Condition 3. Formulas for the Formal Inverse 4. Ideal Membership Results References 1. The Jacobian Conjecture 1.1. The General Assertion. The Jacobian Conjecture can be stated as follows: CO JECTURE 1.1 (JC). For any integer n~1 and polynomials F l ,. .. , F n E qx1, ... , X n ], the polynomial map F = (F l , ... , F n) : en-t en is an automorphism if the determinant jJFI of the Jacobian matrix JF = (DiF j) is a nonzero constant. Here and throughout this paper we write D i for 8j8X i. We will continue to write J F for the Jacobian matrix of a polynomial map F, and the determinant of this matrix will be denoted by IJFI. 1.2. Specific Assertions for Fixed Degree and Dimension. A number of reductions and partial solutions of the problem lead us to formulate the following more specific statements. Note that under the hypothesis of each of these conjectures the conditon "IJFI is a nonzero constant" is equivalent to IJFI = 1 (This can be seen by evaluating at the origin). The following definitions will be useful in stating the conjectures in the section. DEFINITION 1.2. By the degree of a polynomial map F = (F l , ... , F n) we mean the maximum of total degrees of the coordinate functions F l , ... , F n in the variables

Polynomial maps with strongly nilpotent Jacobian matrix and the Jacobian conjecture

Linear Algebra and its Applications, 1996

Let H:k" -+ k" be a polynomial map, It is shown that the Jacobian matrix JH is strongly nilpotent if and only if JH is linearly triangularizable if and only if the polynomial map F = X + H is linearly triangularizable. Furthermore it is shown that for such maps F, sF is linearizable for almost all s E k (except a finite number of roots of unity).

Nilpotent symmetric Jacobian matrices and the Jacobian conjecture

Journal of Pure and Applied Algebra, 2004

It is shown that the Jacobian Conjecture holds for all polynomial maps F : k n → k n of the form F = x + H , such that JH is nilpotent and symmetric, when n 4. If H is also homogeneous a similar result is proved for all n 5.

The Jacobian Conjecture: Linear triangularization for homogeneous polynomial maps in dimension three

Journal of Algebra, 2005

Let k be a field of characteristic zero and F : k 3 → k 3 a polynomial map of the form F = x + H , where H is homogeneous of degree d 2. We show that the Jacobian Conjecture is true for such mappings. More precisely, we show that if J H is nilpotent there exists an invertible linear map T such that T −1 H T = (0, h 2 (x 1 ), h 3 (x 1 , x 2 )), where the h i are homogeneous of degree d. As a consequence of this result, we show that all generalized Drużkowski mappings F = x + H = (x 1 + L d 1 , . . . , x n + L d n ), where L i are linear forms for all i and d 2, are linearly triangularizable if J H is nilpotent and rk J H 3. Bondt), essen@math.ru.nl (A. van den Essen).

On the Jacobian Conjecture

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].