  We provide simple criteria and algorithms for expressing homogeneous
polynomials as sums of powers of independent linear forms, or equivalently, for
decomposing symmetric tensors into sums of rank-1 symmetric tensors of linearly
independent vectors. The criteria rely on two facets of higher degree forms,
namely Harrison's algebraic theory and some algebro-geometric properties. The
proposed algorithms are elementary and based purely on solving linear and
quadratic equations. Moreover, as a byproduct of our criteria and algorithms
one can easily decide whether or not a homogeneous polynomial or symmetric
tensor is orthogonally or unitarily decomposable.
