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Hermitian complexity of real polynomial ideals. (English) Zbl 1273.14120

Summary: We define the Hermitian complexity of a real polynomial ideal and of a real algebraic subset of \(\mathbb C^{n}\). 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.

MSC:

14P05 Real algebraic sets
15B57 Hermitian, skew-Hermitian, and related matrices
32V15 CR manifolds as boundaries of domains
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