Isotropic Rank of Harmonic Polynomials

S. Canino; C. Flavi

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arXiv · 2512.05195

Isotropic Rank of Harmonic Polynomials

S. Canino·C. Flavi

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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.

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S. Canino, C. Flavi. 2025-12-04. Isotropic Rank of Harmonic Polynomials. https://arxiv.org/abs/2512.05195

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