Upper bounds for the spectral norm of symmetric tensors

The maximum of the absolute value of a real homogeneous polynomial of degree d ≥ 3 on the unit sphere corresponds to the spectral norm of the induced real d-symmetric tensor S . We give two sequences of upper bounds on the spectral norm of 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. 2020 Mathematics Subject Classification 15A42, 15A60, 15A69, 15B48.