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Interpolating varieties and counting functions in \(\mathbb{C}^ n\). (English) Zbl 0852.32009

Let a plurisubharmonic function \(p : \mathbb{C}^n \to [0, \infty)\) be a weight function, i.e. \(p\) satisfies the three conditions: \(\log (1 + |z |^2) = O(p(z))\), \(p (|z |) = p(z)\) and \(p(2z) = O(p(z))\). Let \(A(V)\) be the ring of all analytic functions on a subset \(V\) of \(\mathbb{C}^n\). Put \[ A_p (V) : = \biggl \{f \in A(V); \bigl |f(z) \bigr |\leq A \exp \bigl( Bp(z) \bigr) \quad \text{for some} \quad A,B > 0 \biggr\} \] and \[ A^0_p (V) : = \Bigl \{f \in A(V); \forall_{\varepsilon > 0} \sup_{z \in \mathbb{C}^n} \bigl |f(z) \bigr |e^{- \varepsilon p(z)} < \infty \Bigr\}. \] If the restriction map \(\rho : A_p (\mathbb{C}^n) \ni f \to f |V \in A_p (V)\) is onto, then the set \(V\) is called an interpolating variety for \(A_p (\mathbb{C}^n)\). The authors study geometrical conditions on \(V : = f^{-1} (0)\), where \(f \in A_p (\mathbb{C}^n)\) (resp. \(f \in A^0_p (\mathbb{C}^n))\), in order that \(V\) be an interpolating variety for the corresponding ring. The conditions (which in the case of \(n > 1\) appear only sufficient) are expressed in terms of the counting function \(N(r,z,V)\) of \(V\).
Reviewer: J.Siciak (Kraków)

MSC:

32D15 Continuation of analytic objects in several complex variables
32A38 Algebras of holomorphic functions of several complex variables
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