A refinement on the local cactus rank algorithm

TL;DR

This paper introduces an improved algorithm for recovering minimal local apolar schemes to homogeneous polynomials, utilizing Hankel operators to efficiently compute the Hilbert function and distinguish between GAD and extension cases.

Contribution

It provides a new refined algorithm with constructive procedures for identifying minimal local apolar schemes and computing their Hilbert functions efficiently.

Findings

Algorithm successfully recovers minimal local apolar schemes.

Efficient computation of Hilbert functions via Hankel operators.

Distinguishes between GAD and extension cases based on socle degree.

Abstract

We present an algorithm to recover a minimal local apolar scheme to a homogeneous polynomial $F$. The socle degree of the scheme determines whether it is evinced by a Generalized Additive Decomposition (GAD) of $F$ or of an extension. We give constructive procedures for both cases and compute the Hilbert function efficiently via Hankel operators.