Apolarity and direct sum decomposability of polynomials

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.