Sums of squares II: matrix functions

TL;DR

This paper establishes sharp conditions on positive semidefinite matrix functions for their representation as finite sums of squares of vector fields, advancing understanding in hypoellipticity and degenerate regimes.

Contribution

It provides precise criteria for matrix functions to be expressed as sums of squares, extending previous results to more general decompositions involving quasiconformal terms.

Findings

Characterization of matrix functions as sums of squares under specific conditions

Conditions for the determinant vanishing only at the origin

Extension to decompositions with quasiconformal terms

Abstract

This is the second in a series of three papers dealing with sums of squares and hypoellipticity in the infinitely degenerate regime. We give sharp conditions on the entries of a positive semidefinite NxN matrix function F on n-dimensional Euclidean space, whose determinant vanishes only at the origin and such that F is comparable to its diagonal matrix, in order that F is a finite sum of squares of C^2,delta vector fields. We also consider slightly more general decompositions in which a single quasiconformal term need not be a sum of squares.