Spectral norm of a symmetric tensor and its computation

We show that the spectral norm of a d-mode real or complex symmetric tensor in n variables can be computed by finding the fixed points of the corresponding polynomial map. For a generic complex symmetric tensor the number of fixed points is finite, and we give upper and lower bounds for the number of fixed points. For n=2 we show that these fixed points are the roots of a corresponding univariate polynomial of degree at most (d-1)2+1, except certain cases, which are completely analyzed. In particular, for n=2 the spectral norm of d-symmetric tensor is polynomially computable in d with a given relative precision. For a fixed n>2 we show that the spectral norm of a d-mode symmetric tensor is polynomially computable in d with a given relative precision with respect to the Hilbert-Schmidt norm of the tensor. These results show that the geometric measure of entanglement of d-mode symmetric qunits on mathbbCn are polynomially computable for a fixed n.