  The maximum of the absolute value of a real homogeneous polynomial of degree
$d\ge 3$ on the unit sphere corresponds to the spectral norm of the induced
real $d$-symmetric tensor $\mathcal{S}$. We give two sequences of upper bounds
on the spectral norm of $\mathcal{S}$, which are stated in terms of certain
roots of the Hilbert-Schmidt norms of corresponding iterates. We show that
these sequences are converging to a limit, which is the minimal value of these
upper bounds. Some generalizations to iterates of homogeneous polynomial maps
are discussed.
