Direct Sum Decomposability of Polynomials and Factorization of Associated Forms

For a homogeneous polynomial with a non-zero discriminant, we interpret direct sum decomposability of the polynomial in terms of factorization properties of the Macaulay inverse system of its Milnor algebra. This leads to an if-and-only-if criterion for direct sum decomposability of such a polynomial, and to an algorithm for computing direct sum decompositions over any field, either of characteristic 0 or of sufficiently large positive characteristic, for which polynomial factorization algorithms exist. We also give simple necessary criteria for direct sum decomposability of arbitrary homogeneous polynomials over arbitrary fields and apply them to prove that many interesting classes of homogeneous polynomials are not direct sums.