Some remarks on non-symmetric polarization

TL;DR

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Contribution

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Abstract

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 Schl\"uters 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 Schl\"uters' 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.