A family of explicit Waring decompositions of a polynomial

TL;DR

This paper introduces a family of explicit Waring decompositions for any monomial, providing practical tools for polynomial decomposition with applications in computational algebra and tensor analysis.

Contribution

It presents a novel explicit construction for Waring decompositions of monomials, improving bounds on Waring rank and enabling explicit decompositions of general polynomials.

Findings

Provides an explicit formula for Waring decompositions of monomials.

Offers an upper bound for the Waring rank of monomials.

Includes a computer implementation for practical use.

Abstract

In this paper we settle some polynomial identity which provides a family of explicit Waring decompositions of any monomial $X_0^{a_0}X_1^{a_1}\cdots X_n^{a_n}$ over a field $\Bbbk$. This gives an upper bound for the Waring rank of a given monomial and naturally leads to an explicit Waring decomposition of any homogeneous form and, eventually, of any polynomial via (de)homogenization. Note that such decomposition is very useful in many applications dealing with polynomial computations, symmetric tensor problems and so on. We discuss some computational aspect of our result as comparing with other known methods and also present a computer implementation for potential use in the end.