Monomials as sums of powers: the Real binary case

TL;DR

This paper proves that in the real binary case, monomials of degree d that are not powers of a variable require at least d powers of linear forms to be expressed, generalizing Sylvester's example.

Contribution

It extends Sylvester's classical example by establishing a lower bound on the number of powers needed to express certain monomials in two variables.

Findings

Monomials not powers of a variable need at least d powers of linear forms.

The result generalizes Sylvester's example to a broader class of monomials.

Provides a lower bound on the Waring rank for these monomials.

Abstract

We generalize an example, due to Sylvester, and prove that any monomial of degree $d$ in $\mathbb R[x_0, x_1]$, which is not a power of a variable, cannot be written as a linear combination of fewer than $d$ powers of linear forms.