Abstract
We pose and discuss several Hermitian analogues of Hilbert's 17-th problem. We survey what is known, offer many explicit examples and some proofs, and give applications to CR geometry. We prove one new algebraic theorem: a non-negative Hermitian symmetric polynomial divides a non-zero squared norm if and only if it is a quotient of squared norms. We also discuss a new example of Putinar–Scheiderer.
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