Ellipsoidal designs and the Prouhet--Tarry--Escott problem
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| Format: | Preprint |
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2025
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| _version_ | 1866916712017821696 |
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| author | Matsumura, Hideki Sawa, Masanori |
| author_facet | Matsumura, Hideki Sawa, Masanori |
| contents | The notion of ellipsoidal design was first introduced by Pandey (2022) as a full generalization of spherical designs on the unit circle $S^1$. In this paper, we elucidate the advantages of examining the connections between ellipsoidal design and the two-dimensional Prouhet--Tarry--Escott problem, say ${\mathrm PTE}_2$, originally introduced by Alpers and Tijdeman (2007) as a natural generalization of the classical one-dimensional PTE problem (${\mathrm PTE}_1$). We first provide a combinatorial criterion for the construction of solutions of ${\mathrm PTE}_2$ from a pair of ellipsoidal designs. We also give an arithmetic proof of the Stroud-type bound for ellipsoidal designs, and then establish a classification theorem for designs with equality. Such a classification result is closely related to an open question on the existence of rational spherical $4$-designs on $S^1$, discussed in Cui, Xia and Xiang (2019). As far as the authors know, a solution found by Alpers and Tijdeman is the first and the only known parametric ideal solution of degree $5$ for ${\mathrm PTE}_2$. Moreover, as one of our main theorems, we prove that the Alpers--Tijdeman solution is equivalent to a certain two-dimensional extension of the famous Borwein solution for ${\mathrm PTE}_1$. As a by-product of this theorem, we discover a family of ellipsoidal $5$-designs among the Alpers--Tijdeman solution. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2502_17106 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Ellipsoidal designs and the Prouhet--Tarry--Escott problem Matsumura, Hideki Sawa, Masanori Combinatorics Number Theory 05E99, 11D72 (Primary) 05B30, 14G05 (Secondary) The notion of ellipsoidal design was first introduced by Pandey (2022) as a full generalization of spherical designs on the unit circle $S^1$. In this paper, we elucidate the advantages of examining the connections between ellipsoidal design and the two-dimensional Prouhet--Tarry--Escott problem, say ${\mathrm PTE}_2$, originally introduced by Alpers and Tijdeman (2007) as a natural generalization of the classical one-dimensional PTE problem (${\mathrm PTE}_1$). We first provide a combinatorial criterion for the construction of solutions of ${\mathrm PTE}_2$ from a pair of ellipsoidal designs. We also give an arithmetic proof of the Stroud-type bound for ellipsoidal designs, and then establish a classification theorem for designs with equality. Such a classification result is closely related to an open question on the existence of rational spherical $4$-designs on $S^1$, discussed in Cui, Xia and Xiang (2019). As far as the authors know, a solution found by Alpers and Tijdeman is the first and the only known parametric ideal solution of degree $5$ for ${\mathrm PTE}_2$. Moreover, as one of our main theorems, we prove that the Alpers--Tijdeman solution is equivalent to a certain two-dimensional extension of the famous Borwein solution for ${\mathrm PTE}_1$. As a by-product of this theorem, we discover a family of ellipsoidal $5$-designs among the Alpers--Tijdeman solution. |
| title | Ellipsoidal designs and the Prouhet--Tarry--Escott problem |
| topic | Combinatorics Number Theory 05E99, 11D72 (Primary) 05B30, 14G05 (Secondary) |
| url | https://arxiv.org/abs/2502.17106 |