Rank decomposition and symmetric rank decomposition over arbitrary fields

We give a necessary and sufficient condition for symmetric tensors to have symmetric rank decompositions and give some further results on the Comon's Conjecture (i.e. the rank and symmetric rank agree for a symmetric tensor). In particular, as an application, we prove that if an m-order 2-dimensional symmetric tensor has a symmetric rank decomposition, then its symmetric rank equals its rank or its rank plus one.