## Discrete interpolating varieties in pseudoconvex open sets of $$\mathbb C^n$$.(English)Zbl 1118.32013

Let $$\Omega$$ be an open set in $${\mathbb C}^n$$, and $$A(\Omega )$$ the ring of holomorphic functions in $$\Omega$$. The Hörmander algebra $$A_p(\Omega )$$ associated with a plurisubharmonic function $$p:\Omega\to [0,\infty )$$ (a Hörmander weight) is defined as $$A_p(\Omega)=\{ f\in A(\Omega ):\,\exists A,B>0:\;| f(z)| \leq Ae^{Bp(z)},\;z\in\Omega \}$$, and the weight satisfies the following conditions: (i) all polynomials belong to $$A_p(\Omega )$$; (ii) there exist constants $$K_1,\ldots, K_4$$ such that $$z\in\Omega$$ and $$| \zeta -z| \leq e^{-K_1p(z)-K_2}$$ implies that $$\zeta\in\Omega$$ and $$p(\zeta)\leq K_3p(z)+K_4$$.
The author proves that a necessary and sufficient condition for a discrete variety $$V$$ in a pseudoconvex open set $$\Omega\subset {\mathbb C}^n$$ to be an interpolating variety for $$A_p(\Omega)$$ is that there exists a holomorphic mapping $$f:\Omega\to {\mathbb C}^n$$ whose zero set contains $$V$$ and whose Jacobian at the points $$\zeta\in V$$ is bounded from below by $$\epsilon e^{-Cp(\zeta)}$$, for some constants $$\epsilon, C>0$$.

### MSC:

 32E30 Holomorphic, polynomial and rational approximation, and interpolation in several complex variables; Runge pairs 32A15 Entire functions of several complex variables 32C25 Analytic subsets and submanifolds 46E10 Topological linear spaces of continuous, differentiable or analytic functions
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