The Waring Problem of Complex Binary Forms

TL;DR

This paper addresses the Waring problem for complex binary forms, providing elementary solutions and foundational tools like the Apolarity Lemma and Sylvester's Theorem to facilitate future research.

Contribution

It offers an accessible, elementary approach to solving the Waring problem for complex binary forms, laying groundwork for extending to more general cases.

Findings

Solved the Waring problem for complex binary forms

Presented elementary proofs of key algebraic tools

Established a foundation for future generalizations

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.