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Determinantal computation of minimal local GADs

Oriol Reig Fité, Daniele Taufer

TL;DR

This work proposes a determinantal method for computing minimal local GADs by minimizing the rank of a symbolic inverse system and proves that this finiteness is guaranteed whenever the local GAD-rank of the form does not exceed its degree.

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.

Determinantal computation of minimal local GADs

TL;DR

This work proposes a determinantal method for computing minimal local GADs by minimizing the rank of a symbolic inverse system and proves that this finiteness is guaranteed whenever the local GAD-rank of the form does not exceed its degree.

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.
Paper Structure (14 sections, 6 theorems, 49 equations, 1 figure, 1 table, 1 algorithm)

This paper contains 14 sections, 6 theorems, 49 equations, 1 figure, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notation
  4. GADs and natural apolar schemes
  5. Local GADs and inverse systems
  6. Minimal local GAD
  7. Symbolic inverse system
  8. Minimal local GAD algorithm
  9. Generic forms
  10. Rank minimization
  11. Comparison with previous methods
  12. Extension-based algorithms
  13. Catalecticant minors
  14. Conclusions and future directions

Key Result

Proposition 1

Let $\omega \in \mathcal{S}_k$ and $F = \omega x_0^{d-k} \in \mathcal{S}_d$. With the above notation, we have

Figures (1)

  • Figure 1: Different methods for extracting minors from $\mathcal{I}_{F,\ell}(\gamma)$: selecting random minors (A), constructing block-diagonal minors (B), and following contraction chains (C).

Theorems & Definitions (32)

  • Example 1
  • Example 2
  • Proposition 1
  • proof
  • Remark 1
  • Lemma 1
  • proof
  • Example 3
  • Definition 1
  • Remark 2
  • ...and 22 more