On schemes evinced by generalized additive decompositions and their regularity

We define and explicitly construct schemes evinced by generalized additive decompositions (GADs) of a given d-homogeneous polynomial F. We employ GADs to investigate the regularity of 0-dimensional schemes apolar to F, focusing on those satisfying some minimality conditions. We show that irredundant schemes to F need not be d-regular, unless they are evinced by special GADs of F. Instead, we prove that tangential decompositions of minimal length are always d-regular, as well as irredundant apolar schemes of length at most 2d + 1.