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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
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