The Waring Problem of Harmonic Polynomials

TL;DR

This paper explores the Waring problem for harmonic polynomials, establishing that their Waring rank equals their degree, and provides an algorithm for minimal decompositions.

Contribution

It characterizes the annihilating ideal of harmonic polynomials and shows that any linear form can appear in their minimal Waring decompositions.

Findings

Waring rank of harmonic polynomials equals their degree

Any linear form can appear in minimal Waring decompositions

Provides an explicit algorithm for computing decompositions

Abstract

This paper investigates the Waring problem of harmonic polynomials. By characterizing the annihilating ideal of a homogeneous harmonic polynomial, i.e., a real binary form that is in the kernel of the Laplacian, we show that its Waring rank equals its degree. Moreover, we show that any linear form can appear in a minimal Waring decomposition of a homogeneous harmonic polynomial, implying that the forbidden locus is empty. We also provide an explicit algorithm for computing the minimal Waring decompositions.