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The Waring Problem of Harmonic Polynomials

Hua-Lin Huang, Yilun Tang, Yu Ye, Rongmin Zhu

TL;DR

The paper addresses the real Waring problem for homogeneous binary harmonic polynomials of degree $d$ by developing a self-contained real Apolarity framework. It proves the annihilating ideal of a nonzero harmonic form is generated by $\Delta$ and a linear operator $\nabla$, and establishes the exact Waring rank $WR(f)=d$, with every linear form able to appear in a minimal decomposition. A constructive algorithm is given to compute minimal decompositions, and the results extend to forms annihilated by quadratic differential operators, fully solving that subclass. The work also notes that harmonic forms have only real roots, providing explicit decompositions and practical computational methods with potential applications in real algebraic geometry and invariant theory.

Abstract

This paper investigates the Waring problem of harmonic polynomials. By characterizing the annihilating ideal of a homogeneous harmonic polynomial, i.e., a real binary form that is in the kernel of the Laplacian, we show that its Waring rank equals its degree. Moreover, we show that any linear form can appear in a minimal Waring decomposition of a homogeneous harmonic polynomial, implying that the forbidden locus is empty. We also provide an explicit algorithm for computing the minimal Waring decompositions.

The Waring Problem of Harmonic Polynomials

TL;DR

The paper addresses the real Waring problem for homogeneous binary harmonic polynomials of degree by developing a self-contained real Apolarity framework. It proves the annihilating ideal of a nonzero harmonic form is generated by and a linear operator , and establishes the exact Waring rank , with every linear form able to appear in a minimal decomposition. A constructive algorithm is given to compute minimal decompositions, and the results extend to forms annihilated by quadratic differential operators, fully solving that subclass. The work also notes that harmonic forms have only real roots, providing explicit decompositions and practical computational methods with potential applications in real algebraic geometry and invariant theory.

Abstract

This paper investigates the Waring problem of harmonic polynomials. By characterizing the annihilating ideal of a homogeneous harmonic polynomial, i.e., a real binary form that is in the kernel of the Laplacian, we show that its Waring rank equals its degree. Moreover, we show that any linear form can appear in a minimal Waring decomposition of a homogeneous harmonic polynomial, implying that the forbidden locus is empty. We also provide an explicit algorithm for computing the minimal Waring decompositions.
Paper Structure (14 sections, 8 theorems, 24 equations, 1 algorithm)

This paper contains 14 sections, 8 theorems, 24 equations, 1 algorithm.

Table of Contents

  1. Introduction
  2. Problem Statement
  3. Related Work
  4. Methods and Results
  5. Organization of the Paper
  6. Preliminaries
  7. Existence of Waring Decompositions
  8. The Apolarity Lemma
  9. The Waring Problem of Harmonic Polynomials
  10. Properties of Harmonic Forms
  11. Annihilating Ideals of Harmonic forms
  12. The Waring Decompositions of Harmonic forms
  13. Forms Annihilated by Quadratic Differential Operators
  14. Some Examples

Key Result

Lemma 2.1

Let $\alpha_1x+\beta_1y, \ \dots, \ \alpha_{d+1}x+\beta_{d+1}y$ be $d+1$ pairwise non-proportional linear forms. Then their $d$-th powers form a basis of $\mathcal{R}_d$. In particular, every $f \in \mathcal{R}_d$ can be expressed as for some real numbers $\lambda_1, \dots, \lambda_{d+1}$. That is, $f$ admits a Waring decomposition.

Theorems & Definitions (20)

  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Corollary 2.3
  • Example 3.1
  • Proposition 3.2
  • proof
  • Remark 3.3
  • Proposition 3.4
  • ...and 10 more