  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.
