Abstract
We define the Hermitian complexity of a real polynomial ideal and of a real algebraic subset of Cn. This concept is aimed at determining precise necessary conditions for a Hermitian symmetric polynomial to agree with a Hermitian squared norm on an algebraic set. The latter topic has been a central theme in modern polynomial optimization and in complex geometry, specifically related to the holomorphic embedding of pseudoconvex domain into balls, or the classification of proper holomorphic maps between balls.
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