Asymptotic estimates on the von Neumann inequality for homogeneous polynomials

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 $$\mathrm{\parallel }p(T_1,\dots ,T_n){\mathrm{\parallel }}_{\mathcal{L}(\mathcal{H})}\le C_{k,q}(n)\sup \{\mathrm{\mid }p(z_1,\dots ,z_n)\mathrm{\mid }:{\sum }_{i=1}^n\mathrm{\mid }{z}_i{\mathrm{\mid }}^q\le 1\}\mathrm{.}$$ 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 $q=\mathrm{\infty }$, we obtain the correct asymptotic behavior of this constant (answering a question posed by Dixon in the seventies). For $2\le q<\mathrm{\infty }$ 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.