  Our work is related to problems $73$ and $74$ of Mazur and Orlicz in ``The
Scottish Book" (ed. R. D. Mauldin). Let $k_1, \ldots, k_n$ be nonnegative
integers such that $\sum_{i=1}^{n} k_{i}=m$, and let $\mathbb{K}(k_1, \ldots,
k_n; X)$, where $\mathbb{K}=\mathbb{R}$ or $\mathbb{C}$, be the smallest number
satisfying the property: if $L$ is any symmetric $m$-linear form on a Banach
space $X$, then \[ \sup_{\|x_{i}\|\leq 1 ,\atop i=1,2,\ldots ,n}
|L(x_{1}^{k_1},\ldots ,x_{n}^{k_n})|\leq \mathbb{K}(k_1, \ldots, k_n;
X)\sup_{\|x\|\leq 1} |L(x, \ldots ,x)|\,, \] where the exponents $k_{1},
\ldots, k_{n}$, are as described above, and each $k_{i}$ denotes the number of
coordinates in which the corresponding base variable appears. In the case of
complex Banach spaces, the problem of optimising the constant $\mathbb{C}(k_1,
\ldots, k_n; X)$ is well-studied. In the more challenging case of real Banach
spaces much less is known about the estimates for $\mathbb{R}(k_1, \ldots, k_n;
X)$. In this work, both real and complex settings are examined using results
from the local theory of Banach spaces, as well as from interpolation theory of
linear operators. In the particular case of complex $L^{p}(\mu)$ spaces, and
for certain values of $p$, our results are optimal. As an application, we prove
Markov-type inequalities for homogeneous polynomials on Banach spaces.
