 
Summary: Matrix Representations for Positive
Noncommutative Polynomials
J.William Helton
Scott A. McCullough
Mihai Putinar
September 17, 2003
Abstract
In real semialgebraic geometry it is common to represent a polyno
mial q which is positive on a region R as a weighted sum of squares.
Serious obstructions arise when q is not strictly positive on the re
gion R. Here we are concerned with noncommutative polynomials and
obtaining a representation for them which is valid even when strict
positivity fails.
Specifically, we treat a "symmetric" polynomial q(x, h) in noncom
muting variables, {x1, . . . , xgx
} and {h1, . . . , hgh
} for which q(X, H) is
positive semidefinite whenever
X = (X1, . . . , Xgx
) and H = (H1, . . . , Hgh
