A note on the simultaneous Waring rank of monomials

TL;DR

This paper investigates the complex simultaneous Waring rank of collections of monomials, providing bounds and formulas, with applications to binomials and maximal ranks, supported by algebraic and combinatorial methods and computational tools.

Contribution

It introduces new bounds and formulas for the simultaneous Waring rank of monomials, advancing understanding in algebraic complexity and tensor decomposition.

Findings

Provided a lower bound for general collections of monomials

Derived a formula for special collections' simultaneous Waring rank

Included Macaulay2 scripts for experimental verification

Abstract

In this paper we study the complex simultaneous Waring rank for collections of monomials. For general collections we provide a lower bound, whereas for special collections we provide a formula for the simultaneous Waring rank. Our approach is algebraic and combinatorial. We give an application to ranks of binomials and maximal simultaneous ranks. Moreover, we include an appendix of scripts written in the algebra software Macaulay2 to experiment with simultaneous ranks.