From Apolar Schemes to Generalized Affine Decompositions

In many cases classical catalecticants fail to reveal the apolar scheme of a form. In this talk we show how common eigenvectors and higher-order generalized eigenvectors encode the geometry of apolar schemes, leading from Waring to tangential decompositions and ultimately to generalized affine decompositions (GAfD).
We present examples where the local apolar structure is too large to admit any generalized additive decomposition (GAdD), although the GAfD is fully reconstructible, and discuss the implications for decomposition algorithms and rank notions.

This seminar is part of the Problems in Algebraic Geometry Coming from Complexity Theory series.