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Non-symmetric polarization

Let $P$ be an $m$-homogeneous polynomial in $n$-complex variables $x_1, \dotsc, x_n$. Clearly, $P$ has a unique representation in the form \begin{equation*} P(x)= \sum_{1 \leq j_1 \leq \dotsc \leq j_m \leq n} c_{(j_1, \dotsc, j_m)} \, x_{j_1} \dotsb x_{j_m} \,, \end{equation*} and the $m$"~form \begin{equation*} L_P(x^{(1)}, \dotsc, x^{(m)})= \sum_{1 \leq j_1 \leq \dotsc \leq j_m \leq n} c_{(j_1, \dotsc, j_m)} \, x^{(1)}_{j_1} \dotsb x^{(m)}_{j_m} \end{equation*} satisfies $L_P(x,\dotsc, x) = P(x)$ for every $x\in\mathbb{C}^n$. We show that, although $L_P$ in general is non-symmetric, for a large class of reasonable norms $ \lVert \cdot \rVert $ on $\mathbb{C}^n$ the norm of $L_P$ on $(\mathbb{C}^n, \lVert \cdot \rVert )^m$ up to a logarithmic term $(c \log n)^{m^2}$ can be estimated by the norm of $P$ on $ (\mathbb{C}^n, \lVert \cdot \rVert )$; here $c \ge 1$ denotes a universal constant. Moreover, for the $\ell_p$"~norms $ \lVert \cdot \rVert_p$, $1 \leq p < 2$ the logarithmic term in the number $n$ of variables is even superfluous.