Monomials as sum of k-th powers of forms

TL;DR

This paper investigates expressing monomials as sums of k-th powers of forms, providing bounds and exact solutions for specific cases, advancing understanding in polynomial Waring problems.

Contribution

It introduces a general bound on the number of summands for representing monomials as sums of powers, with complete solutions for the k=3 case in two and three variables.

Findings

Established a general bound on summands for monomials in polynomial rings.

Refined bounds specifically for two-variable cases.

Solved completely the k=3 case for monomials in two and three variables.

Abstract

Motivated by recent results on the Waring problem for polynomial rings and representation of monomial as sum of powers of linear forms, we consider the problem of presenting monomials of degree kd as sums of k-th powers of forms of degree d. We produce a general bound on the number of summands for any number of variables which we refine in the two variables case. We completely solve the k=3 case for monomials in two and three variables.