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Many interesting non-linear functions of several variables like $\|x\|^2 =\sum\limits_{k=1}^n x_k^2$ become linear in "different variables". Is there a characterization of all (continuous, differentiable,...) functions $f:\mathbb R^n \to \mathbb R^m$ such that there exist $h:\mathbb R\to\mathbb R$ and a linear map $A:\mathbb R^n\to \mathbb R^m$ with $f(x_1,\ldots,x_n)= A(h(x_1),\ldots,h(x_n))$ for all $x=(x_1,\ldots,x_n)\in \mathbb R^n$?

This smells like a functional equation but knowing close to nothing about this I dare to ask this question here.

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It seems to me that you are willing for the implicit function theorem and the Weiestrass preparation lemma

See

http://en.wikipedia.org/wiki/Implicit_function_theorem

http://en.wikipedia.org/wiki/Weierstrass_preparation_theorem

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    $\begingroup$ This might be related but I don't see how this answers the question. In particular, I asked for a global representation whereas the proosed tools are local. $\endgroup$ – Jochen Mar 25 '14 at 5:41
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    $\begingroup$ I see, I misunderstud a little the question, I was thinking that $h$ is a diffeo of $\mathbb R^n$, while you allow only diffeos of the form $\otimes^n h$ $\endgroup$ – user126154 Mar 25 '14 at 10:19

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