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. © 2005 Elsevier Inc. All rights reserved.