Some remarks on non-symmetric polarization

Abstract

Let $P:{\mathbb{C}}^n\to \mathbb{C}$ 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 $L_P:{\left({\mathbb{C}}^n\right)}^m\to \mathbb{C}$ 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 $({\mathbb{C}}^n,\Vert \cdot \Vert {)}^m$ by the norm of $P$ on $({\mathbb{C}}^n,\Vert \cdot \Vert )$ times a $(c\log n{)}^{m^2}$ factor for every 1-unconditional norm $\Vert \cdot \Vert$ 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 $(cm\log n{)}^{m-1}$ 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 $\Vert \cdot {\Vert }_p$ with $1\le p<2$ is addressed.