Determinantal computation of minimal local GADs

TL;DR

This paper introduces a determinantal approach for computing minimal local generalized additive decompositions (GADs) of homogeneous polynomials, providing a practical method that is independent of apolarity action and effective when the minimal support locus is finite.

Contribution

It proposes a novel determinantal method for minimal local GAD computation based on rank minimization of symbolic inverse systems, applicable without tensor extensions.

Findings

The method is effective when the minimal support locus is finite.

Finiteness of the locus is guaranteed when the local GAD-rank does not exceed the degree.

Computational experiments compare the approach with existing algorithms.

Abstract

We study local generalized additive decompositions (GADs) of homogeneous polynomials and their associated point schemes through their local inverse systems. We prove that their construction and algebraic properties are independent of the chosen apolarity action. We propose a determinantal method for computing minimal local GADs by minimizing the rank of a symbolic inverse system. When the locus of minimal supports is finite, this provides a practical method to determine all minimal local decompositions without tensor extensions. We prove that this finiteness is guaranteed whenever the local GAD-rank of the form does not exceed its degree. We analyze both generic and special cases, provide computational evidence assessing the impact of different choices for minors, and compare our approach with existing algorithms for local apolar schemes.