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Let $A_{p,r}^m(n)$ be the best constant that fulfills the followinginequality: for every $m$-homogeneous polynomial $P(z) = \sum_{|\alpha|=m}a_{\alpha} z^{\alpha}$ in $n$ complex variables, $$\big( \sum_{|\alpha|=m}|a_{\alpha}|^{r} \big)^{1/r} \leq A_{p,r}^m(n) \sup_{z \in B_{\ell_p^n}}\big|P(z) \big| .$$ For every degree $m$, and a wide range of values of $ p,r\in [1,\infty]$ (including any $r$ in the case $p \in [1,2]$, and any $r$ and$p$ for the 2-homogeneous case), we give the correct asymptotic behavior ofthese constants as $n$ (the number of variables) tends to infinity. Remarkably,in many cases, extremal polynomials for these inequalities are not (astraditionally expected) found using classical random unimodular polynomials,and special combinatorial configurations of monomials are needed. Namely, weshow that Steiner polynomials (i.e., $m$-homogeneous polynomials such that themulti-indices corresponding to the nonzero coefficients form partial Steinersystems), do the work for certain range of values of $p,r$.  As a byproduct, we present some applications of these estimates to theinterpolation of tensor products of Banach spaces, to the study of (mixed)unconditionality in spaces of polynomials and to the multivariable vonNeumann's inequality.

The sup-norm vs. the norm of the coefficients: equivalence constants for homogeneous polynomials