A new bound for the Real Waring rank of monomials

TL;DR

This paper introduces a new upper bound for the real and rational Waring rank of monomials, improving upon existing bounds and matching known cases, with practical examples and computational methods.

Contribution

It provides a novel structured apolar set approach that yields a tighter upper bound for the Waring rank of monomials over reals and rationals.

Findings

New upper bound matches known real rank cases for n=1 and min(a_i)=1

Bound is lower than previous known bounds for real Waring rank

Method applicable over rational numbers with computational examples

Abstract

In this paper we consider the Waring rank of monomials over the real and the rational numbers. We give a new upper bound for it by establishing a way in which one can take a structured apolar set for any given monomial $X_0^{a_0}X_1^{a_1}\cdots X_n^{a_n}$ ($a_i>0$). This bound coincides with the real Waring rank in the case $n=1$ and in the case $\min(a_i)=1$, which are all the known cases for the real rank of monomials. Our bound is also lower than any other known general bounds for the real Waring rank. Since all of the constructions are still valid over the rational numbers, this provides a new result for the rational Waring rank of any monomial as well. Some examples and computational implementation for potential use are presented in the end.