I hope to show:

any complex homogeneous bivariate polynomial of degree d is the sum
of d terms of the form (ax+by)^{d}, and there exist polynomials for
which one really needs d such terms; and

any complex nxnmatrix with trace zero and an n in the upper right
corner is the sum of n rank one matrices, each having trace zero and
a 1 in the upper right corner;
and to explain the broader problem of which these are special
cases. `Hope', for our proof of the second part is not yet quite
finished. `Our': Karin Baur's and mine.
