  In what follows, we pose two general conjectures about decompositions of
homogeneous polynomials as sums of powers. The first one (suggested by G.
Ottaviani) deals with the generic k-rank of complex-valued forms of any degree
divisible by k in any number of variables. The second one (by the fourth
author) deals with the maximal k-rank of binary forms. We settle the first
conjecture in the cases of two variables and the second in the first
non-trivial case of the 3-rd powers of quadratic binary forms.
