Isotropic Rank of Harmonic Polynomials
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866912750046806016 |
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| author | Canino, S. Flavi, C. |
| author_facet | Canino, S. Flavi, C. |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_05195 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Isotropic Rank of Harmonic Polynomials Canino, S. Flavi, C. Algebraic Geometry Commutative Algebra 14N05, 14N07, 14N15, 15A21 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. |
| title | Isotropic Rank of Harmonic Polynomials |
| topic | Algebraic Geometry Commutative Algebra 14N05, 14N07, 14N15, 15A21 |
| url | https://arxiv.org/abs/2512.05195 |