  We consider the nonconvex optimization problem associated with the
decomposition of a real symmetric tensor into a sum of rank one terms. Use is
made of the rich symmetry structure to construct infinite families of critical
points represented by Puiseux series in the problem dimension, and so obtain
precise analytic estimates on the value of the objective function and the
Hessian spectrum. The results allow an analytic characterization of various
obstructions to using local optimization methods, revealing in particular a
complex array of saddles and minima differing by their symmetry, structure and
analytic properties. A~desirable phenomenon, occurring for all critical points
considered, concerns the number of negative Hessian eigenvalues increasing with
the value of the objective function. Our approach makes use of Newton polyhedra
as well as results from real algebraic geometry, notably the Curve Selection
Lemma, to determine the extremal character of degenerate critical points,
establishing in particular the existence of infinite families of third-order
saddles which can significantly slow down the optimization process.
