Recent progress on the Jacobian Conjecture (original) (raw)
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A note on the Jacobian Conjecture
Linear Algebra and its Applications, 2011
In this note, we show that, if the Druzkowski mappings F (X) = X + (AX) * 3 , i.e. F (X) = (x 1 + (a 11 x 1 + • • • + a 1n x n) 3 , • • • , x n + (a n1 x 1 + • • • + a nn x n) 3), satisfies T rJ((AX) * 3) = 0, then rank(A) ≤ 1 2 (n + δ) where δ is the number of diagonal elements of A which are equal to zero. Furthermore, we show the Jacobian Conjecture is true for the Druzkowski mappings in dimension ≤ 9 in the case n i=1 a ii = 0.
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].
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
Some remarks on the Jacobian question
Proceedings Mathematical Sciences, 1994
This revised version of Abhyankar's old lecture notes contains the original proof of the Galois case of the n-variable Jacobian problem. They also contain proofs for some cases of the 2-variable Jacobian, including the two characteristic pairs case. In addition, proofs of some of the well-known formulas enunciated by Abhyankar are actually written down. These include the Taylor Resultant Formula and the Semigroup Conductor formula for plane curves. The notes are also meant to provide inspiration for applying the expansion theoretic techniques to the Jacobian problem.
On the Jacobian conjecture in two variables
Journal of Pure and Applied Algebra, 1988
The user has requested enhancement of the downloaded file. ON THE JACOBIAN CONJECTURE IN TWO VARIABLES Let k he a Ii&l of characteristic zcru The two-dimensional Jawhian conjecture state< that given f ;lnd x in k[x. ~1, if the Jacobian nf (.f. g) 1s %I nun-zurcl cunstimt, then k[f. g] x: k1.r. ~1. In this paper WC Eivc nuw proofs of known cquivalznt versions of this wnjccture. Proposition 1.1 (Appelgate and Onishi [2, $121). Let (f, g) be a basic pair. Let deg( f) = dm > 1, deg( g) = dn > 1, where gcd(m, n) = 1. Then for each direction ( p, q) there is a ( p, q)-form h of positive degree such that h" -ff,, and h" -g;.,. 0