This is the first in a series of three papers dealing with sums of squares
and hypoellipticity in the infinite regime. We give a sharp sufficient
condition on a smooth nonnegative function f on n-dimensional Euclidean space
so that it can be written as a finite sum of squares of C^2,delta functions.
Special consideration is given to analyzing the case when f vanishes only at
the origin, answering a question of Bony et al.