Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Can anyone give a nonlinear regular function from C^2 to C^2 with a constant nonzero Jacobian? It seems to me that the only such functions are linear.

According to the Jacobian conjecture, a function from C^2 to C^2 with a constant nonzero Jacobian must have an inverse.

share|improve this question
    
You apparently will have to clarify: did you mean a constant Jacobian matrix or a constant Jacobian determinant? Those two are related, but different. –  J. M. Dec 15 '10 at 1:33
    
I meant Jacobian determinant. –  Craig Feinstein Dec 15 '10 at 1:35

3 Answers 3

up vote 3 down vote accepted

Some trial and error should get it.

I figured $\left[\begin{array}{cc} x & 1\\ x-1 & 1\end{array}\right]$ was a good target Jacobian matrix, and so got $f(x,y)=(\frac 1 2 x^2 + y,\frac 1 2 x^2-x + y)$. Is that what you're looking for? Evidently it has an inverse, $g(a,b)=(a-b,a-\frac 1 2(a-b)^2)$.

share|improve this answer
    
Yes, that's what I am looking for. Evidently, I was wrong when I thought they couldn't exist. Now, I'll ask what is the most general form? –  Craig Feinstein Dec 15 '10 at 1:38

If $u(x,y)$ is of the form $af(x-y)+bx$, where $a$ and $b$ are constants and $f$ is a one-variable differentiable function, and if $v(x,y)=y-x$, then $$ \frac{{\partial u(x,y)}}{{\partial x}} \frac{{\partial v(x,y)}}{{\partial y}} - \frac{{\partial u(x,y)}}{{\partial y}}\frac{{\partial v(x,y)}}{{\partial x}} = [af'(x-y)+b]+[-af'(x-y)] = b. $$

EDIT: More simply, letting $u(x,y)=af(x+y)+bx$ and $v(x,y)=x+y$ gives the same result as above.

share|improve this answer

Any linear function has constant Jacobian determinant, as does any map of the form $(z_1,z_2) \rightarrow (z_1, z_2 - f(z_1))$ or $(z_1,z_2) \rightarrow (z_1 - f(z_2), z_2)$. As a result, any finite composition of maps of these forms will have constant Jacobian. These include the examples that Shai Covo and Andrew Marshall listed. I forget if there are known examples outside this category.

share|improve this answer

Your Answer

 
discard

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

Not the answer you're looking for? Browse other questions tagged or ask your own question.