  A polynomial is a direct sum if it can be written as a sum of two non-zero
polynomials in some distinct sets of variables, up to a linear change of
variables. We analyze criteria for a homogeneous polynomial to be decomposable
as a direct sum, in terms of the apolar ideal of the polynomial. We prove that
the apolar ideal of a polynomial of degree $d$ strictly depending on all
variables has a minimal generator of degree $d$ if and only if it is a limit of
direct sums.
