Waring-like decompositions of polynomials - 1

TL;DR

This paper explores a generalized Waring decomposition for homogeneous polynomials, expressing them as sums of monomials evaluated at linear forms, extending classical polynomial decomposition methods.

Contribution

It introduces a new framework for decomposing polynomials into sums of monomials evaluated at linear forms, broadening the scope of traditional Waring decompositions.

Findings

Developed a generalized decomposition method for homogeneous forms.

Provided conditions for the existence of such decompositions.

Illustrated the approach with specific examples.

Abstract

Let $F$ be a homogeneous form of degree $d$ in $n$ variables. A Waring decomposition of $F$ is a way to express $F$ as a sum of $d^{th}$ powers of linear forms. In this paper we consider the decompositions of a form as a sum of expressions, each of which is a fixed monomial evaluated at linear forms.