Structures of the Length Seven Power Sum Decompositions of Ternary Quartics

The problem of finding the maximum symmetric rank in dimension three—which, to our knowledge, is known only for degrees up to five—aims at a deeper understanding of tensor rank, especially in view of its computational complexity applications. We test a conjectural answer by pursuing a strategy that has led to the best known upper bound to date. As a first outcome, our main result reveals a connection between Waring decompositions and certain spectral data. We then highlight some examples of quartics that exhibit a special behavior. Finally, our findings indicate that a different conjectural answer, presented here for the first time, may be more plausible.