  By the von Neumann inequality for homogeneous polynomials there exists a
positive constant $C_{k,q}(n)$ such that for every $k$-homogeneous polynomial
$p$ in $n$ variables and every $n$-tuple of commuting operators $(T_1, \dots,
T_n)$ with $\sum_{i=1}^{n} \Vert T_{i} \Vert^{q} \leq 1$ we have \[ \|p(T_1,
\dots, T_n)\|_{\mathcal L(\mathcal H)} \leq C_{k,q}(n) \; \sup\{ |p(z_1, \dots,
z_n)| : \textstyle \sum_{i=1}^{n} \vert z_{i} \vert^{q} \leq 1 \}\,. \] For
fixed $k$ and $q$, we study the asymptotic growth of the smallest constant
$C_{k,q}(n)$ as $n$ (the number of variables/operators) tends to infinity. For
$q = \infty$, we obtain the correct asymptotic behavior of this constant
(answering a question posed by Dixon in the seventies). For $2 \leq q < \infty$
we improve some lower bounds given by Mantero and Tonge, and prove the
asymptotic behavior up to a logarithmic factor. To achieve this we provide
estimates of the norm of homogeneous unimodular Steiner polynomials, i.e.
polynomials such that the multi-indices corresponding to the nonzero
coefficients form partial Steiner systems.
