  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 $\mathbb{C}^n$ are polynomially computable for a fixed $n$.
