Geometric designs and Hilbert-Kamke equations of degree five for classical orthogonal polynomials
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| Format: | Preprint |
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2025
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| author | Mishima, Teruyuki Lu, Xiao-Nan Sawa, Masanori Uchida, Yukihiro |
| author_facet | Mishima, Teruyuki Lu, Xiao-Nan Sawa, Masanori Uchida, Yukihiro |
| contents | In this paper we elucidate the advantage of examining the connections between Hilbert-Kamke equations and geometric designs, or Chebyshev-type quadrature, for classical orthogonal polynomials. We first establish that if a $5$-design with $6$ rational points for a symmetric classical measure is parametrized by rational functions, then the corresponding measure should be the Chebyshev measure $(1-t^2)^{-1/2}dt/π$ on $(-1,1)$. Our proof is based on the collaboration of a certain polynomial identity and some advanced techniques on the computation of the genus of a certain irreducible curve. Next, we prove a necessary and sufficient condition for the existence of rational $5$-designs for the Chebyshev measure. Moreover, as one of our main theorems, we construct an infinite family of ideal solutions for the Prouhet-Tarry-Escott (PTE) problem by utilizing rational $5$-designs for the Chebyshev measure, and then establish that, up to affine equivalence over $\mathbb{Q}$, such ideal solutions are included in the famous parametric solutions found by Borwein (2002). |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2503_21151 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Geometric designs and Hilbert-Kamke equations of degree five for classical orthogonal polynomials Mishima, Teruyuki Lu, Xiao-Nan Sawa, Masanori Uchida, Yukihiro Number Theory Combinatorics 33C45, 05E99, 65D32 (Primary) 12E10, 11E76 (Secondary) In this paper we elucidate the advantage of examining the connections between Hilbert-Kamke equations and geometric designs, or Chebyshev-type quadrature, for classical orthogonal polynomials. We first establish that if a $5$-design with $6$ rational points for a symmetric classical measure is parametrized by rational functions, then the corresponding measure should be the Chebyshev measure $(1-t^2)^{-1/2}dt/π$ on $(-1,1)$. Our proof is based on the collaboration of a certain polynomial identity and some advanced techniques on the computation of the genus of a certain irreducible curve. Next, we prove a necessary and sufficient condition for the existence of rational $5$-designs for the Chebyshev measure. Moreover, as one of our main theorems, we construct an infinite family of ideal solutions for the Prouhet-Tarry-Escott (PTE) problem by utilizing rational $5$-designs for the Chebyshev measure, and then establish that, up to affine equivalence over $\mathbb{Q}$, such ideal solutions are included in the famous parametric solutions found by Borwein (2002). |
| title | Geometric designs and Hilbert-Kamke equations of degree five for classical orthogonal polynomials |
| topic | Number Theory Combinatorics 33C45, 05E99, 65D32 (Primary) 12E10, 11E76 (Secondary) |
| url | https://arxiv.org/abs/2503.21151 |