For which $n$, $f:(x_1,\dots,x_n)\mapsto(x_1x_2^2,x_2x_3^2,\dots,x_{n-1}x_n^2,x_nx_1^2)$ is diffeomorphism?

Question

I've had following problem to solve on exam and I did not know how to do it.

For which $n>2$, $f$ is diffeomorphism? $$ f:\mathbb{R}_+^n\rightarrow\mathbb{R}_+^n $$ $$ f:(x_1,\dots,x_n)\mapsto(x_1x_2^2,x_2x_3^2,\dots,x_{n-1}x_n^2,x_nx_1^2) $$ Where $\mathbb{R}_+=(0,\infty)$

Answer 1

Hint. Your function is the composition of

• Taking the logarithm at each coordinate
• Applying a certain linear transformation
• Taking the antilogarithm at each coordinate

So you have a series of mappings $\mathbb R_+^n\to \mathbb R^n \to \mathbb R^n \to \mathbb R_+^n $. The first and last are clearly diffeomorphisms. How about the middle one?