On Waring rank jumps via critical rank-one approximations
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866914534316310528 |
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| author | Oneto, Alessandro Santarsiero, Pierpaola Turatti, Ettore Teixeira |
| author_facet | Oneto, Alessandro Santarsiero, Pierpaola Turatti, Ettore Teixeira |
| contents | We investigate whether eigenvectors, also known as critical rank-one approximations, of a symmetric tensor can be used to increase or decrease its Waring rank. First, we study the variety of degree-d rank-r forms which admit an eigenvector as part of a minimal Waring decomposition. In the case of binary forms, we show that this is of codimension-one in the r-th secant variety of the rational normal curve. On the other hand, we prove that for any binary form of rank less than (d+1)/2 (subgeneric), any eigenvector increases the rank. Additionally, when the degree is odd, the same holds for generic forms of generic rank. Our approach employs the strict relation between the apolar action and the Bombieri-Weyl product. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2605_04695 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On Waring rank jumps via critical rank-one approximations Oneto, Alessandro Santarsiero, Pierpaola Turatti, Ettore Teixeira Algebraic Geometry 14N07, 15A18, 15A69 We investigate whether eigenvectors, also known as critical rank-one approximations, of a symmetric tensor can be used to increase or decrease its Waring rank. First, we study the variety of degree-d rank-r forms which admit an eigenvector as part of a minimal Waring decomposition. In the case of binary forms, we show that this is of codimension-one in the r-th secant variety of the rational normal curve. On the other hand, we prove that for any binary form of rank less than (d+1)/2 (subgeneric), any eigenvector increases the rank. Additionally, when the degree is odd, the same holds for generic forms of generic rank. Our approach employs the strict relation between the apolar action and the Bombieri-Weyl product. |
| title | On Waring rank jumps via critical rank-one approximations |
| topic | Algebraic Geometry 14N07, 15A18, 15A69 |
| url | https://arxiv.org/abs/2605.04695 |