Sets non-thin at \(\infty \) in \(\mathbb C^m\) (Q1025029)

Let \(E\) be a closed subset of \(\mathbb C^m.\) The author proves the following two results. Theorem 2. \(E\) is non-thin at \(\infty\) iff \[ \limsup_{n \to \infty} |P_n (z)|^{1/k_n} \leq 1,\;\forall z \in E, \] implies the correctness of this inequality for all \(z \in \mathbb C^m.\) Here, \(\{P_n\}\) is a sequence of polynomials and \(k_n \geq \deg P_n.\) Theorem 3. The pluricomplex Green function \(V_E (z)\) of the set \(E\) with pole at infinity is equal to zero identically iff any open neighborhood of \(E\) is non-thin at \(\infty.\)