Nonnegative functions as squares or sums of squares

We prove that, for n?4, there are C^∞ nonnegative functions f of n variables (and even flat ones for n?5) which are not a finite sum of squares of C² functions. For n=1, where a decomposition in a sum of two squares is always possible, we investigate the possibility of writing f=g². We prove that, in general, one cannot require a better regularity than g∈C¹. Assuming that f vanishes at all its local minima, we prove that it is possible to get g∈C² but that one cannot require any additional regularity.