A constructive approach to the truncated moment problem on cubic curves in Weierstrass form
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866918492372992000 |
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| author | Bhardwaj, Abhishek Zalar, Aljaž |
| author_facet | Bhardwaj, Abhishek Zalar, Aljaž |
| contents | In this paper, we develop a constructive solution for the pure truncated moment problem on cubic curves in Weierstrass form, establishing the existence of a representing measure whose number of atoms equals the rank of the associated moment matrix. By a recent result of Baldi, Blekherman, and Sinn, for projectively smooth curves whose projective closure has exactly one real point at infinity, the existence of such a rank-attaining atomic measure is equivalent to the existence of a representing measure; consequently, the TMP is constructively solved for this class of curves. We also present a numerical degree--$6$ example in which every minimal representing measure supported on the cubic curve requires $\operatorname{rank} M(3)+1$ atoms, where $M(3)$ denotes the moment matrix. Finally, we provide a constructive solution for the symmetric case, i.e., when all moments of odd degree in $y$ vanish. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_08719 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A constructive approach to the truncated moment problem on cubic curves in Weierstrass form Bhardwaj, Abhishek Zalar, Aljaž Functional Analysis Primary 44A60, 47A57, 47A20, Secondary 47N40 In this paper, we develop a constructive solution for the pure truncated moment problem on cubic curves in Weierstrass form, establishing the existence of a representing measure whose number of atoms equals the rank of the associated moment matrix. By a recent result of Baldi, Blekherman, and Sinn, for projectively smooth curves whose projective closure has exactly one real point at infinity, the existence of such a rank-attaining atomic measure is equivalent to the existence of a representing measure; consequently, the TMP is constructively solved for this class of curves. We also present a numerical degree--$6$ example in which every minimal representing measure supported on the cubic curve requires $\operatorname{rank} M(3)+1$ atoms, where $M(3)$ denotes the moment matrix. Finally, we provide a constructive solution for the symmetric case, i.e., when all moments of odd degree in $y$ vanish. |
| title | A constructive approach to the truncated moment problem on cubic curves in Weierstrass form |
| topic | Functional Analysis Primary 44A60, 47A57, 47A20, Secondary 47N40 |
| url | https://arxiv.org/abs/2605.08719 |