Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

To make myself precise, I would like to recall some backgrounds.

(Markus-Yamabe, $\mathrm{MY}_n$) Given a $C^1$ map $f:\Bbb R^n \to \Bbb R^n$ with $f(0)=0$ and $Df$ everywhere Hurwitz stable (the real part of all eigenvalues of $Df$ are $<0$), 0 is the global attractor of dynamical system $\dot{x}=f(x)$.

$\mathrm{MY}_2$ is now a theorem (Fessler; Glutsyuk; Gutierrez), and a polynomial counterexample has been found for $\mathrm{MY}_3$ (Cima et al.)

I would also like to distinguish 2 kinds of "real Jacobian conjecture":

(real Jacobian, $\mathrm{RJ}_n$) Given a polynomial map $f:\Bbb R^n \to \Bbb R^n$, the Jacobian $J_f$ being some non-zero constant implies $f$ is a diffeomorphism. I believe a stronger statement can be made by taking $f$ to be $C^1$.

(strong Jacobian, $\mathrm{SJ}_n$) Given a $C^1$ map $f:\Bbb R^n \to \Bbb R^n$, $J_f(x)>0$ implies $f$ is a diffeomorphism.

Notice that Pinchuk has found a polynomial counterexample for $\mathrm{SJ}_2$.

I learnt recently a very vague statement that the Markus-Yamabe conjecture implies the real Jacobian conjecture. I think these stuffs must be well-known among the experts, so could anyone

(1)make this statement rigorous by indicating the relationship between $\mathrm{MY}_n$, $\mathrm{RJ}_n$ and $\mathrm{SJ}_n$?

(2)or even better, show me how the argument goes,

(3)or locate some reference?

Thanks a lot!

share|improve this question
    
Have you looked at the book `Polynomial Automorphism' by Arno Van den Essen? He certainly mentions several things about M-Y conjecture. –  Mohan Aug 22 '12 at 19:18
add comment

Know someone who can answer? Share a link to this question via email, Google+, Twitter, or Facebook.

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Browse other questions tagged or ask your own question.