Isotropic Rank of Harmonic Polynomials

TL;DR

This paper introduces the concept of isotropic rank for harmonic polynomials, providing a full classification of its values across dimensions and specific classes, extending the Alexander-Hirschowitz theorem.

Contribution

It determines the isotropic rank for general harmonic forms and classifies it for specific cases, establishing a comprehensive framework similar to the Alexander-Hirschowitz theorem.

Findings

Classified the isotropic rank for general harmonic forms.

Solved the isotropic rank problem for ternary forms, quadrics, and monomials.

Established the dimensions of secant varieties related to isotropic linear forms.

Abstract

Any homogeneous harmonic polynomial can be decomposed as a sum of powers of isotropic linear forms, that is, linear forms whose coefficients are the coordinates of isotropic points. The minimum size of such decompositions for a harmonic polynomial is called its isotropic rank. As with the Waring rank, the problem of determining the isotropic rank of a given harmonic form is very hard. We determine the isotropic rank of a general harmonic form providing a full classification of the dimensions of secant varieties of the variety of d-powers of isotropic linear forms in n+1 variables, for every n,d, thus obtaining the analogue of the widely-celebrated Alexander-Hirschowitz theorem. Moreover, we completely solve the problem of determining the isotropic rank for the following classes of harmonic forms: ternary forms, quadrics and monomials.