Upper bounds for the spectral norm of symmetric tensors

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.