  At the time of writing, the general problem of finding the maximal Waring
rank for homogeneous polynomials of fixed degree and number of variables (or,
equivalently, the maximal symmetric rank for symmetric tensors of fixed order
and in 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 maximal rank for plane quartics
is seven, following the elementary geometric idea of splitting power sum
decompositions along three suitable lines.
