Nonnegative functions as squares or sums of squares

It is proven that, for $n\geq 4$, there are $C^{\infty}$ nonnegative functions $f$ of $n$ variables (and even flat ones for $n\geq 5$) which are not a finite sum of squares of $C^{2}$ functions. For $n=1$, where a decomposition in a sum of two squares is always possible, the possibility of writing $f=g^{2}$ is investigated. We prove that, in general, one cannot require a better regularity than $g\in C^{1}$. Assuming that $f$ vanishes at all its local minima, it is proved that it is possible to get $g\in C^{2}$ but that one cannot require any additional regularity.