A proof that the maximum rank for ternary quartics is seven

SUMMARY

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.