Higher Order Polars and Dual Forms

Abstract

We review the classical theory of apolarity and investigate its applications in relation to power sum decompositions. Higher order polars admits, in a natural way, a duality between graded symmetric algebras. This duality can be expressed via a matrix called the catalecticant and we present its close relation to the Waring rank. Finite, zero-dimensional schemes corresponding to Artinian Gorenstein rings are studied, and techniques for finding so-called apolar schemes are presented. For any homogeneous form of even degree one can construct a dual form via apolarity. We investigate how such forms behave in relation to their dual forms. We look at apolar schemes and present precise criteria for determining when the catalecticant and cactus rank for a ternary homogeneous form differ. Lastly, we develop a method for computing explicit power sum decompositions of ternary homogeneous forms of even degree.