A proof that the maximum rank for ternary quartics is seven

At the time of writing, the general problem of finding the maximum Waring rank for homogeneous polynomials of a fixed degree and in a fixed number of variables (or, equivalently, the maximum symmetric rank for symmetric tensors of a fixed order and in a fixed dimension) is still unsolved. To our knowledge, the answer for ternary quartics is not widely known and can only be found among the results of a master's thesis by Johannes Kleppe at the University of Oslo (1999). In the present work we give a (direct) proof that the maximum rank for plane quartics is seven, following the elementary geometric idea of splitting power sum decompositions along three suitable lines.