  We use an algebraic approach to construct minimal decompositions of symmetric
tensors with low rank. This is done by using Apolarity Theory and by studying
minimal sets of reduced points apolar to a given symmetric tensor, namely,
whose ideal is contained in the apolar ideal associated to the tensor. In
particular, we focus on the structure of the Hilbert function of these ideals
of points. We give a procedure which produces a minimal set of points apolar to
any symmetric tensor of rank at most 5. This procedure is also implemented in
the algebra software Macaulay2.
