Some remarks on non-symmetric polarization

Let $P:{\mathbb{C}}^n\to \mathbb{C}$ be an $m$-homogeneous polynomial given by $$P(x)=\sum _{1\le j_1\le \dots \le j_m\le n}{c}_{j_1\dots j_m}{x}_{j_1}\dots x_{j_m}\mathrm{.}$$ Defant and Schl"uters defined a non-symmetric associated $m$-form $L_P:{\left({\mathbb{C}}^n\right)}^m\to \mathbb{C}$ by $$L_P(x^{(1)},\dots ,x^{(m)})=\sum _{1\le j_1\le \dots \le j_m\le n}{c}_{j_1\dots j_m}{x}_{j_1}^{(1)}\dots x_{j_m}^{(m)}\mathrm{.}$$ They estimated the norm of $L_P$ on $({\mathbb{C}}^n,\mathrm{\parallel }\cdot \mathrm{\parallel }{)}^m$ by the norm of $P$ on $({\mathbb{C}}^n,\mathrm{\parallel }\cdot \mathrm{\parallel })$ times a $(c\log n{)}^{m^2}$ factor for every 1-unconditional norm $\mathrm{\parallel }\cdot \mathrm{\parallel }$ 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 $(cm\log n{)}^{m-1}$ is provided. Regarding the lower bound, it is shown that the optimal constant is bigger than $(c\log n{)}^{m\mathrm{/}2}$ when $n\gg m$. Finally, the case of ${\mathrm{\ell }}_p$-norms $\mathrm{\parallel }\cdot {\mathrm{\parallel }}_p$ with $1\le p<2$ is addressed.