Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Isotropic Rank of Harmonic Polynomials

S. Canino, C. Flavi

TL;DR

This work introduces the isotropic rank for homogeneous harmonic polynomials, defining decompositions into sums of $d$-th powers of isotropic linear forms and proving an Alexander–Hirschowitz–type classification for the dimensions of secant varieties of isotropic Veronese varieties. It develops an algebro-geometric framework via apolarity theory adapted to harmonic polynomials, and uses the differential Horace method to solve interpolation problems that determine secant dimensions in all degrees. The authors obtain the isotropic rank of a generic harmonic form and provide complete results for binary, ternary, quadratic, and monomial cases, along with a detailed analysis of the quadratic (harmonic) forms through canonical block decompositions. The results reveal sharp bounds relating Waring and isotropic ranks and yield effective, constructive decompositions in several important families, highlighting a deep link between representation theory and tensor decomposition in the harmonic setting. The work thus extends classical rank theory to the harmonic context, with potential implications for invariant theory, spherical harmonics, and geometric complexity in tensor problems.

Abstract

Any homogeneous harmonic polynomial can be decomposed as a sum of powers of isotropic linear forms, that is, linear forms whose coefficients are the coordinates of isotropic points. The minimum size of such decompositions for a harmonic polynomial is called its isotropic rank. As with the Waring rank, the problem of determining the isotropic rank of a given harmonic form is very hard. We determine the isotropic rank of a general harmonic form providing a full classification of the dimensions of secant varieties of the variety of d-powers of isotropic linear forms in n+1 variables, for every n,d, thus obtaining the analogue of the widely-celebrated Alexander-Hirschowitz theorem. Moreover, we completely solve the problem of determining the isotropic rank for the following classes of harmonic forms: ternary forms, quadrics and monomials.

Isotropic Rank of Harmonic Polynomials

TL;DR

This work introduces the isotropic rank for homogeneous harmonic polynomials, defining decompositions into sums of -th powers of isotropic linear forms and proving an Alexander–Hirschowitz–type classification for the dimensions of secant varieties of isotropic Veronese varieties. It develops an algebro-geometric framework via apolarity theory adapted to harmonic polynomials, and uses the differential Horace method to solve interpolation problems that determine secant dimensions in all degrees. The authors obtain the isotropic rank of a generic harmonic form and provide complete results for binary, ternary, quadratic, and monomial cases, along with a detailed analysis of the quadratic (harmonic) forms through canonical block decompositions. The results reveal sharp bounds relating Waring and isotropic ranks and yield effective, constructive decompositions in several important families, highlighting a deep link between representation theory and tensor decomposition in the harmonic setting. The work thus extends classical rank theory to the harmonic context, with potential implications for invariant theory, spherical harmonics, and geometric complexity in tensor problems.

Abstract

Any homogeneous harmonic polynomial can be decomposed as a sum of powers of isotropic linear forms, that is, linear forms whose coefficients are the coordinates of isotropic points. The minimum size of such decompositions for a harmonic polynomial is called its isotropic rank. As with the Waring rank, the problem of determining the isotropic rank of a given harmonic form is very hard. We determine the isotropic rank of a general harmonic form providing a full classification of the dimensions of secant varieties of the variety of d-powers of isotropic linear forms in n+1 variables, for every n,d, thus obtaining the analogue of the widely-celebrated Alexander-Hirschowitz theorem. Moreover, we completely solve the problem of determining the isotropic rank for the following classes of harmonic forms: ternary forms, quadrics and monomials.

Paper Structure

This paper contains 17 sections, 52 theorems, 257 equations.

Table of Contents

  1. Introduction
  2. Harmonic polynomials and their applications
  3. Isotropic rank
  4. Apolarity Theory and harmonic polynomials
  5. Apolarity
  6. Harmonic forms
  7. Isotropic rank of harmonic polynomials
  8. Secant varieties of isotropic Veronese varieties
  9. The case d=2
  10. The translation of the problem
  11. The differential Horace method
  12. The solution of the interpolation problem in degree 3
  13. The solution of the interpolation problem for dgeq4
  14. Binary and ternary harmonic forms
  15. Harmonic quadratic forms
  16. ...and 2 more sections

Key Result

Theorem 1

Let $n,d,r\in\mathbb{N}_{\geq 1}$. If $d\geq 3$, then while, if $d=2$, then In particular, the isotropic rank of a generic harmonic polynomial of $\mathcal{H}_{n,2}$ is $n+1$, and the isotropic rank of a generic harmonic polynomial of $\mathcal{H}_{n,d}$, with $d\geq 3$, is

Theorems & Definitions (104)

  • Theorem 1
  • Proposition 2
  • Theorem 3
  • Theorem 4
  • Theorem 5
  • Definition 2.2
  • Definition 2.3
  • Lemma 6: Ger96*Proposition 2.5 and Proposition 2.6
  • Lemma 7: Apolarity lemma IK99*Lemma 1.15
  • Definition 2.4
  • ...and 94 more