Sums of hermitian squares on pseudoconvex boundaries

We give an abstract characterization of all real algebraic subvarieties of complex affine space on which every positive polynomial is a sum of hermitian squares, and we find obstructions to this phenomenon. As a consequence we construct a strictly pseudoconvex domain with smooth algebraic boundary on which there exists a degree two positive polynomial which is not a sum of hermitian squares, answering thus in the negative a question of John D'Angelo.

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Published 1 January 2010