Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Eigenvectors of tensors and algorithms for Waring decomposition

Luke Oeding, Giorgio Ottaviani

TL;DR

The paper develops a constructive program for Waring decompositions of symmetric tensors by combining classical catalecticant methods with Koszul flattenings and a bundle-theoretic framework. It introduces a general algorithm based on eigenvectors of tensors and a Koszul-flattening map $P_f$, then extends this with a bundle-based approach that uses a vector bundle $E$ on a projective variety to reconstruct decompositions from the kernel of a linear map $A_f$, producing explicit results for quintics and the Sylvester pentahedral case. Counts of eigenvectors via Chern classes (Cartwright–Sturmfels formulas) support a deeper geometric understanding of the decomposition problem, and the Macaulay2 implementations demonstrate practical computability. Overall, the work provides a unified, geometry-informed toolkit for symmetric tensor decomposition with concrete rank bounds and verifiable examples, along with accessible software to reproduce and extend the results.

Abstract

A Waring decomposition of a (homogeneous) polynomial f is a minimal sum of powers of linear forms expressing f. Under certain conditions, such a decomposition is unique. We discuss some algorithms to compute the Waring decomposition, which are linked to the equation of certain secant varieties and to eigenvectors of tensors. In particular we explicitly decompose a general cubic polynomial in three variables as the sum of five cubes (Sylvester Pentahedral Theorem).

Eigenvectors of tensors and algorithms for Waring decomposition

TL;DR

The paper develops a constructive program for Waring decompositions of symmetric tensors by combining classical catalecticant methods with Koszul flattenings and a bundle-theoretic framework. It introduces a general algorithm based on eigenvectors of tensors and a Koszul-flattening map , then extends this with a bundle-based approach that uses a vector bundle on a projective variety to reconstruct decompositions from the kernel of a linear map , producing explicit results for quintics and the Sylvester pentahedral case. Counts of eigenvectors via Chern classes (Cartwright–Sturmfels formulas) support a deeper geometric understanding of the decomposition problem, and the Macaulay2 implementations demonstrate practical computability. Overall, the work provides a unified, geometry-informed toolkit for symmetric tensor decomposition with concrete rank bounds and verifiable examples, along with accessible software to reproduce and extend the results.

Abstract

A Waring decomposition of a (homogeneous) polynomial f is a minimal sum of powers of linear forms expressing f. Under certain conditions, such a decomposition is unique. We discuss some algorithms to compute the Waring decomposition, which are linked to the equation of certain secant varieties and to eigenvectors of tensors. In particular we explicitly decompose a general cubic polynomial in three variables as the sum of five cubes (Sylvester Pentahedral Theorem).

Paper Structure

This paper contains 17 sections, 20 theorems, 77 equations.

Table of Contents

  1. Introduction
  2. Classical methods for tensor decomposition: Sylvester's catalecticant method
  3. General results on the symmetric rank
  4. The catalecticant method
  5. Limits of the catalecticant method
  6. New methods for tensor decomposition: Koszul flattening and eigenvectors of tensors
  7. The quotient bundle and eigenvectors of tensors
  8. Pentahedral example
  9. New methods for tensor decomposition: bundle method
  10. The bundle construction
  11. A side remark on presentations
  12. Vector bundles, statement and proofs of the main results
  13. Quintic and Pentahedral examples revisited
  14. Rank bounds for feasibility of our algorithms
  15. Using Chern classes to count the number of eigenvectors of a general tensor
  16. ...and 2 more sections

Key Result

Theorem 2.2

Let $V$ be a complex vector space of dimension $n+1$. The general $f\in S^dV$ has rank which is called the generic rank, with the only exceptions

Theorems & Definitions (46)

  • Remark 1.1
  • Remark 2.1
  • Theorem 2.2: AH2
  • Theorem 2.3: CC02, Mella06, Ballico2005
  • Theorem 2.4
  • proof
  • Theorem 2.5
  • Lemma 3.1
  • proof
  • Definition 3.2
  • ...and 36 more