Waring decompositions of special binomials

TL;DR

This paper determines the Waring rank of specific binomials involving a monomial and a linear form, using algebraic geometry tools, and shows the monomial's minimal decomposition length is exactly $(k+1)^2$.

Contribution

It provides the exact Waring rank for a class of special binomials and reveals the minimal decomposition length of a key monomial, advancing understanding in polynomial decomposition.

Findings

Waring rank of $x^ky^kz^k + ext{linear form}^{3k}$ is determined.

The monomial $x^ky^kz^k$ has no irredundant decompositions of length $(k+1)^2 + 1$.

The study uses Hilbert functions and point configurations in projective space.

Abstract

We determine the Waring rank of homogeneous polynomials of the form $x^ky^kz^k + \ell^{3k}$ where $\ell$ is a linear form. The result is based on the study of the Hilbert function and the resolution of special configurations of points in $\mathbb{P}^2$. As a byproduct of our result, we show that the monomial $x^ky^kz^k$ does not have irredundant decompositions of length $(k+1)^2 +1$.