  Motivated by the search for a deeper understanding of tensor rank, in view of
its computational complexity applications, we investigate a possible path to
determine the maximum symmetric rank in given degree and dimension. We work in
terms of Waring rank of forms, and aiming to set up a firm basis for an
induction procedure we examine some technical tools to organize length seven
Waring decompositions of ternary quartics, that may turn to be fundamental.
