关于非对称偏振的几点注记
Some remarks on non-symmetric polarization
摘要：
设$P:\mathbb{C}^n\rightarrow \mathbb{C}$是由\[P(x)= \sum_{1\leq j_1\leq \ldots \leq j_m \leq n} c_{j_1 \ldots j_m} x_{j_1}\ldots x_{j_m}.\] Defant给出的$m$ -齐次多项式，Schlüters通过\[L_P \left(x^{(1)},\ldots,x^{(m)} \right)= \sum_{1\leq j_1\leq \ldots \leq j_m \leq n} c_{j_1 \ldots j_m} x_{j_1}^{(1)}\ldots x_{j_m}^{(m)}.\]定义了一个非对称的关联$m$ -形式$L_P: \left(\mathbb{C}^n \right)^m\rightarrow \mathbb{C}$，他们通过对$\mathbb{C}^n$上的每个1-范数无条件$\|\cdot\|$，用$P$在$(\mathbb{C}^n, \| \cdot\|)$上的范数乘以一个$(c\log n)^{m^2}$因子来估计$L_P$在$(\mathbb{C}^n, \| \cdot\|)^m$上的范数。基于洗牌算法的对称化过程（连同Defant和Schlüters的论证）将常数项降为$(c m \log n)^{m-1}$。关于下限，当$n\gg m$时，最佳常数大于$(c \log n)^{m/2}$。最后，讨论了$\ell_p$ -范数$\|\cdot \|_p$与$1\leq p <2$的情形.
英文摘要：
Let $P:\mathbb{C}^n\rightarrow \mathbb{C}$ be an $m$-homogeneous polynomial given by \[P(x)= \sum_{1\leq j_1\leq \ldots \leq j_m \leq n} c_{j_1 \ldots j_m} x_{j_1}\ldots x_{j_m}.\] Defant and Schlüters defined a non-symmetric associated $m$-form $L_P: \left(\mathbb{C}^n \right)^m\rightarrow \mathbb{C}$ by \[L_P \left(x^{(1)},\ldots,x^{(m)} \right)= \sum_{1\leq j_1\leq \ldots \leq j_m \leq n} c_{j_1 \ldots j_m} x_{j_1}^{(1)}\ldots x_{j_m}^{(m)}.\] They estimated the norm of $L_P$ on $(\mathbb{C}^n, \| \cdot\|)^m$ by the norm of $P$ on $(\mathbb{C}^n, \| \cdot\|)$ times a $(c\log n)^{m^2}$ factor for every 1-unconditional norm $\|\cdot\|$ on $\mathbb{C}^n$. A symmetrization procedure based on a card-shuffling algorithm which (together with Defant and Schlüters' argument) brings the constant term down to $(c m \log n)^{m-1}$ is provided. Regarding the lower bound, it is shown that the optimal constant is bigger than $(c \log n)^{m/2}$ when $n\gg m$. Finally, the case of $\ell_p$-norms $\|\cdot \|_p$ with $1\leq p <2$ is addressed.
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