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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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Related: math.polytechnique.fr/~bony/BBCP_jfa.pdf –  Benjamin Dickman Aug 31 '12 at 21:11
@Bazin: see the edit to my answer. –  Igor Rivin Sep 1 '12 at 20:05
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1 Answer

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:

Bony, Jean-Michel(F-POLY-CMT) Sommes de carrés de fonctions dérivables. (French. English, French summary) [Sums of squares of derivable functions] Bull. Soc. Math. France 133 (2005), no. 4, 619–639.

For functions from $\mathbb{R}^k \rightarrow \mathbb{R}_+$ these results are extended in:

Nonnegative functions as squares or sums of squares􏰿 Jean-Michel Bonya, Fabrizio Brogliab, Ferruccio Colombinib, Ludovico Pernazzac (J. Func. An, 2006)

EDIT I asked J-M Bony for the scoop, and his response is that no one seems to know what the counterexample actually is, P.J. Cohen was asked about this shortly before his death, but did not remember, Bony himself says he does not even know whether a counterexample exists, and would not know which way to wager. Given that he is THE expert in the field, I would say the problem is open.

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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
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