Nonnegative functions as squares or sums of squares

We prove that, for n >= 4, there are C-infinity nonnegative functions f of n variables (and even flat ones for n >= 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, we investigate the possibility of writing f = g(2). We prove that, in general, one cannot require a better regularity than g is an element of C-1. Assuming that f vanishes at all its local minima, we prove that it is possible to get g is an element of C-2 but that one cannot require any additional regularity. (c) 2005 Elsevier Inc. All rights reserved.