The polarization constant of a Banach space X is defined as \[{\text{c}}(X){\text{ }}{\text{ }}\mathop {\lim }\limits_{k \to \infty } {\text{ }}\sup {\text{c}}{(k,X)^{\frac{1}{k}}},\]where \[{\text{c}}(k,X)\] stands for the best constant \[C > 0\] such that \[\mathop P\limits^ \vee \leqslant CP\] for every k-homogeneous polynomial \[P \in \mathcal{P}{(^k}X)\]. We show that if X is a finite dimensional complex space then \[{\text{c}}(X) = 1\]. We derive some consequences of this fact regarding the convergence of analytic functions on such spaces.The result is no longer true in the real setting. Here we relate this constant with the so-called Bochnak’s complexification procedure.We also study some other properties connected with polarization. Namely, we provide necessary conditions related to the geometry of X for \[c(2,X) = 1\] to hold. Additionally we link polarization constants with certain estimates of the nuclear norm of the product of polynomials.

中文翻译：

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有限维复空间的极化常数为 1

Banach 空间的极化常数X定义为\[{\text{c}}(X){\text{ }}{\text{ }}\mathop {\lim }\limits_{k \to \infty } {\text{ }}\sup {\text {c}}{(k,X)^{\frac{1}{k}}},\]在哪里\[{\text{c}}(k,X)\]代表最佳常数\[C > 0\]这样\[\mathop P\limits^ \vee \leqslant CP\]对于每个ķ-齐次多项式\[P \in \mathcal{P}{(^k}X)\]. 我们证明如果X是一个有限维复空间\[{\text{c}}(X) = 1\]. 关于解析函数在此类空间上的收敛，我们得出了这一事实的一些结果。结果在实际环境中不再正确。在这里，我们将这个常数与所谓的 Bochnak 复化过程联系起来。我们还研究了与极化有关的其他一些性质。即，我们提供与几何形状相关的必要条件X为了\[c(2,X) = 1\]举行。此外，我们将极化常数与多项式乘积的核范数的某些估计联系起来。