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Positivity conditions for Hermitian symmetric functions. (English) Zbl 1116.32001
Let $$M$$ be a complex manifold and let be $$M^\prime$$ its complex conjugated manifold. A holomorphic function $$R$$ on $$M\times M^\prime$$ is called Hermitian symmetric if $$R(z, \overline{ w}) = \overline {R(w \overline{z}) }$$, for all $$z, \;w$$ in $$M$$.
Let $${\mathcal P}_0(M)$$ be the collection of Hermitian symmetric functions on $$M\times M^\prime$$. For $$N$$ a positive integer or infinty, the authors introduce $${\mathcal P}_N(M) \subset {\mathcal P}_0(M)$$, defining $$R \in {\mathcal P}_N(M)$$ if $$\sum_{i, j = 1}^N R(z_i, \overline{z_j}) a_i \overline{a}_j \geq 0$$, for all $$(z_1,\dots, z_N) \in M^N$$ and all $$a\in \mathbb C^N$$. The main results are as follow: $${\mathcal P }_j (\mathbb C^2 ) \neq {\mathcal P}_k (\mathbb C^2)$$ when $$j\neq k$$, a precise description of when a member of a discrete collection of natural one-parameter families of Hermitian symmetric polynomials lies in $${\mathcal P}_k(\mathbb C^2)$$, a subset $$S$$ of Hermitian symmetric functions on $$M\times M'$$ is stable if there is a finite $$\kappa$$ for which $$S\cap {\mathcal P}_\kappa (M) = S \cap {\mathcal P}_\infty(M)$$. The authors show that $$\kappa$$ is related to the number of positive eigenvalues of the underlying matrix of the coefficients of a Hermitian symmetric function.

##### MSC:
 32A10 Holomorphic functions of several complex variables 32J25 Transcendental methods of algebraic geometry (complex-analytic aspects) 32Q40 Embedding theorems for complex manifolds
##### Keywords:
Hermitian symmetric functions; holomorphic functions
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