# Nonnegative smooth function as sum of squares of smooth functions

There is a famous open problem, whose solution is attributed to Paul Cohen, but no published paper seems to be available:

There exists $f\in C^\infty(\mathbb R,\mathbb R_+)$ such that $f$ is not a finite sum of squares of $C^\infty$ functions.

I would be grateful for any hint or reference to that specific question.

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– Benjamin Dickman Aug 31 '12 at 21:11
@Bazin: see the edit to my answer. – Igor Rivin Sep 1 '12 at 20:05

It seems to be a result of J-M Bony that every nonnegative function in $C^{2m}$ is a sum of squares of two $C^m$ functions, which means that every $C^\infty$ function is the sum of squares of two $C^m$ functions for any $m$ (which, I suppose, does not mean that you can do it with two $C^\infty$ functions -- the counterexamples are attributed to Paul Cohen and D.B.A. Epstein -- see references 1and 4 in the cited paper): the reference is:
For functions from $\mathbb{R}^k \rightarrow \mathbb{R}_+$ these results are extended in:
Thanks for these references, which I knew. As you point out, Bony's result with $C^m$ functions does not extend obviously to the case $m=+\infty$. Looking at the details of the proof, it is clear that the decomposition is changing drastically when $m$ increases. My problem is that these references to P. Cohen result are not supported by any real article or preprint. So somehow the core of my question is: Does anybody ever put his hand on a P.Cohen preprint tackling that counterexample of a $C^\infty$ nonnegative function which is not a finite sum of squares of $C^\infty$ functions ? – Bazin Sep 1 '12 at 8:22