Nonnegative functions as squares or sums of squares

Abstract

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 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 ∈ C 1 . Assuming that f vanishes at all its local minima, we prove that it is possible to get g ∈ C 2 but that one cannot require any additional regularity.

A regularity class for the roots of nonnegative functions

“…Theorem 3.5 of Bony et al [1] establishes that this is a necessary and sufficient condition for a four times continuously differentiable function to have a twice continuously differentiable, admissible square root. This is comparable to the flatness seminorm (2.1), which for β = 4 also gives that f should be bounded by f 1/2 .…”

“…This result is sharp in general in the sense that there exist infinitely differentiable functions such that no admissible square root has Hölder index larger than one (Theorem 2.1 in [1]). Flatness conditions thus become necessary for β > 2 in order to ensure that a β-smooth function has an admissible square root with Hölder index greater than one.…”

“…The condition f ∈ F β requires a certain notion of "flatness" on the derivatives of f that is different from that considered in [1][2][3]. In particular, it contains functions not covered by their results for which Theorem 1 nonetheless applies [see (2.2) for a simple example].…”

“…The key property for additional regularity of roots is flatness, in the sense that whenever f is small then so are its derivatives, and in a series of recent papers, Bony et al [1][2][3] consider flatness conditions for admissible square roots. Recall that g is an admissible square root of f if f = g 2 (allowing g to switch signs).…”

“…Suppose f possesses a converging sequence of local minima (x n ) n such that f (x n ) is strictly positive for all n and f (x n ) → 0 as n → ∞. The nonnegative square root is then the most regular admissible square root, and there exist infinitely differentiable functions of this form with no admissible square root having Hölder index larger than one [1]. In this article, we restrict to studying roots of nonnegative functions that do not change sign, which we without loss of generality take to be positive.…”