Strong paucity in the Brüdern-Robert Diophantine system
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866908754067324928 |
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| author | Wooley, Trevor D. |
| author_facet | Wooley, Trevor D. |
| contents | Let $k$ be a natural number with $k\ge 2$, and let $\varepsilon>0$. We consider the number $V_k^*(P)$ of integral solutions of the system of simultaneous Diophantine equations \[ x_1^{2j-1}+\ldots +x_{k+1}^{2j-1}=y_1^{2j-1}+\ldots +y_{k+1}^{2j-1}\quad (1\le j\le k), \] with $1\le x_i,y_i\le P$ $(1\le i\le k+1)$. Writing $L_k^*(P)$ for the number of diagonal solutions with $\{x_1,\ldots ,x_{k+1}\}=\{y_1,\ldots ,y_{k+1}\}$, so that $L_k^*(P)\sim (k+1)!P^{k+1}$, we prove that \[ V_k^*(P)-L_k^*(P)\ll P^{\sqrt{8k+9}-1+\varepsilon}. \] This establishes a strong paucity result improving on earlier work of Brüdern and Robert. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_05121 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Strong paucity in the Brüdern-Robert Diophantine system Wooley, Trevor D. Number Theory 11D45, 11D72, 11L15, 11P05 Let $k$ be a natural number with $k\ge 2$, and let $\varepsilon>0$. We consider the number $V_k^*(P)$ of integral solutions of the system of simultaneous Diophantine equations \[ x_1^{2j-1}+\ldots +x_{k+1}^{2j-1}=y_1^{2j-1}+\ldots +y_{k+1}^{2j-1}\quad (1\le j\le k), \] with $1\le x_i,y_i\le P$ $(1\le i\le k+1)$. Writing $L_k^*(P)$ for the number of diagonal solutions with $\{x_1,\ldots ,x_{k+1}\}=\{y_1,\ldots ,y_{k+1}\}$, so that $L_k^*(P)\sim (k+1)!P^{k+1}$, we prove that \[ V_k^*(P)-L_k^*(P)\ll P^{\sqrt{8k+9}-1+\varepsilon}. \] This establishes a strong paucity result improving on earlier work of Brüdern and Robert. |
| title | Strong paucity in the Brüdern-Robert Diophantine system |
| topic | Number Theory 11D45, 11D72, 11L15, 11P05 |
| url | https://arxiv.org/abs/2601.05121 |