Apolarity and direct sum decomposability of polynomials

Abstract

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.

35 pages. v2: added remarks generalizing results to arbitrary characteristic. v3: extensive rewrite, several generalizations in last section, numerous clarifications throughout. v4: another extensive rewrite with improved notation and terminology, clarifications and corrections throughout, to incorporate suggestions of referee