The Jacobian Conjecture in its simplest form is the following: Jacobian Conjecture for two variables: Given two polynomials f(x,y), g(x,y) in two variables over a field k of characteristic 0, suppose that the following Jacobian condition is satisfied,

2024年2月15日 · This problem is now known as Keller's problem but is more often called the Jacobian conjecture. This conjecture is still open (1999) for all $n \geq 2$. Polynomial mappings satisfying $\operatorname {det} JF \in \mathbf {C}^*$ are called Keller mappings. Various special cases have been proved:

The Jacobian Conjecture gives a condition for when a function from C nto C which restricts to a polynomial function in each coordinate has an inverse which is also given by polynomials.

The Jacobian conjecture Conjecture Let F : Cn!Cn be a polynomial map such that jJ F(z)j2C then F is invertible (and its inverse is a polynomial map). Still open for n 2!!! More generally Instead of C, we can consider any algebraically closed eld k with characteristic 0. Damiano Fulghesu An introduction to the Jacobian conjecture

We show that the Jacobian conjecture can be reduced to a weaker conjecture in which all fibers of coordinate functions are irreducible. 1. Introduction. Let (x, y) be a coordinate system in C2. The Jacobian Conjecture says that. a polynomial mapping (p, q): C2 — C2 whose Jacobian J(p, q) — dp/dx • dq/dy-dp/dy-dq/dx is equal to 1 is invertible.

The Jacobian conjecture in the plane, first stated by Keller (1939), states that given a ring map of (the polynomial ring in two variables over the complex numbers ) to itself that fixes and sends , to , respectively, is an automorphism iff the Jacobian is a nonzero element of . The condition can easily shown to be necessary, but proving ...

2004年11月10日 · The Jacobian Conjecture is one of the most well-known open problems in algebraic geometry. It now seems that a proof has been found by Carolyn Dean of the University of Michigan, for the case of polynomials in two complex variables (for more variables, many people believe it is not even true).

2014年9月14日 · First of all, I'm not quite aware of central problems where the Jacobian Conjecture (JC) is a prerequisite, but there are several related, or even equivalent problems. First of all, there's the cancellation problem: Let $k$ be an algebraically closed field of …

Jacobian Conjecture (for short (JC) n) (JC) n [F2P(Kn) and JacF(x) 6= 0 for every x2Kn] )[F is injective]; and the so called Generalized Jacobian Conjecture (for short: (GJC)), namely (GJC) (JC) n holds for every n>1: If K=C, then we call the Jacobian Conjecture the complex Jacobian Conjecture (resp. the real Jacobian Conjecture if K=R).

1990年5月15日 · The Jacobian conjecture in two variables is studied. It is shown that if ƒ, g∈ C [x,y] have unit Jacobian but C [ƒ,g]≠ C [x,y], then necessarily gcd(deg(ƒ),deg(g))≥16. Other restrictions on counterexamples are also obtained.