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Matrix Representations for Positive Noncommutative Polynomials

Summary: Matrix Representations for Positive
Noncommutative Polynomials
J.William Helton
Scott A. McCullough
Mihai Putinar
September 17, 2003
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


Source: Akhmedov, Azer - Department of Mathematics, University of California at Santa Barbara


Collections: Mathematics