Hermitian analogues of Hilbert's 17-th problem

Details Hermitian analogues of Hilbert's 17-th problem Hermitian analogues of Hilbert's 17-th problem

2010-12-11

arxiv.org/abs/1012.2479v1

Mathematics

Complex Variables

to appear in Advances in Math

Scientific paper

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 nonzero 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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