Maximum Waring ranks of monomials

TL;DR

This paper investigates the Waring ranks of monomials and their sums in multiple variables, revealing they are generally below the generic rank with some exceptions, and provides asymptotic comparisons.

Contribution

It establishes bounds on the Waring ranks of monomials and sums of coprime monomials in four or more variables, identifying cases where ranks are below the generic rank.

Findings

Monomials and sums of coprime monomials have Waring rank less than the generic rank in four or more variables.

Identifies a short list of exceptions where the rank does not follow this pattern.

Provides asymptotic comparisons between these ranks and the generic rank.

Abstract

We show that monomials and sums of pairwise coprime monomials in four or more variables have Waring rank less than the generic rank, with a short list of exceptions. We asymptotically compare their ranks with the generic rank.