Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

The Waring Problem of Complex Binary Forms

Hua-Lin Huang, Haoran Miao, Yu Ye

TL;DR

The paper addresses the Waring problem for complex binary forms by developing an accessible linear-algebraic framework that relies on Vandermonde-based bases, the Apolarity Lemma, and Hankel-matrix eigenstructure. It establishes a basic upper bound WR(f) ≤ d and provides a constructive algorithm to compute the Waring rank and a minimal decomposition, connecting odd and even degree cases through differentiation and integration. The work offers elementary proofs of Sylvester's 1851 theorem for odd degrees and Gundelfinger's extension for even degrees, with a clear interpretation of each summand as an eigenvalue of a key matrix. Together, these results yield a transparent, implementable approach to binary Waring decompositions and point toward extensions to the multivariate setting.

Abstract

The Waring problem of forms concerns the expression of homogeneous multivariate polynomials as sums of powers of linear forms. This paper focuses on complex binary forms, and we solve the Waring problem for them using basic tools in algebra and analysis. In particular, we present elementary treatments of the Apolarity Lemma and Sylvester's 1851 Theorem, which are easily accessible and will provide an ideal approach for future extension to the general case.

The Waring Problem of Complex Binary Forms

TL;DR

The paper addresses the Waring problem for complex binary forms by developing an accessible linear-algebraic framework that relies on Vandermonde-based bases, the Apolarity Lemma, and Hankel-matrix eigenstructure. It establishes a basic upper bound WR(f) ≤ d and provides a constructive algorithm to compute the Waring rank and a minimal decomposition, connecting odd and even degree cases through differentiation and integration. The work offers elementary proofs of Sylvester's 1851 theorem for odd degrees and Gundelfinger's extension for even degrees, with a clear interpretation of each summand as an eigenvalue of a key matrix. Together, these results yield a transparent, implementable approach to binary Waring decompositions and point toward extensions to the multivariate setting.

Abstract

The Waring problem of forms concerns the expression of homogeneous multivariate polynomials as sums of powers of linear forms. This paper focuses on complex binary forms, and we solve the Waring problem for them using basic tools in algebra and analysis. In particular, we present elementary treatments of the Apolarity Lemma and Sylvester's 1851 Theorem, which are easily accessible and will provide an ideal approach for future extension to the general case.

Paper Structure

This paper contains 3 sections, 10 theorems, 71 equations, 1 algorithm.

Table of Contents

  1. Introduction
  2. The Apolarity Lemma
  3. The Waring decompositions

Key Result

Lemma 2.1

Let $\beta_1,\beta_2,\dots,\beta_r\in\mathbb{C}$ be pairwise distinct with $1\leqslant r\leqslant d+1$. Then the $r$ binary forms $(x+\beta_1y)^d,(x+\beta_2y)^d,\dots,(x+\beta_ry)^d$ are linearly independent. In particular, if $r=d+1$, then they constitute a basis of $\mathcal{R}_d$.

Theorems & Definitions (27)

  • Lemma 2.1
  • proof
  • Corollary 2.2: Existence of Waring decompositions of binary forms
  • Remark 2.3
  • Theorem 2.4: Apolarity Lemma
  • proof
  • Remark 2.5
  • Corollary 2.6
  • Corollary 2.7
  • proof
  • ...and 17 more