Sum of squares I: scalar functions

TL;DR

This paper establishes a precise condition under which a smooth, nonnegative function can be expressed as a finite sum of squares of C^2,delta functions, addressing a specific case where the function vanishes only at the origin.

Contribution

It provides a sharp sufficient condition for representing certain nonnegative functions as finite sums of squares, advancing understanding of sums of squares in the context of hypoellipticity.

Findings

Identifies a sharp sufficient condition for sum of squares representation.

Analyzes the case when the function vanishes only at the origin.

Addresses a question posed by Bony et al.

Abstract

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.