Structure and paucity in affine diagonal systems, I
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arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866912871373340672 |
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| author | Brandes, Julia Wooley, Trevor D. |
| author_facet | Brandes, Julia Wooley, Trevor D. |
| contents | Let $\varepsilon>0$ and $\mathbf h\in \mathbb Z^3$. We show that whenever $P$ is large and the system \[ x_1^j+x_2^j-y_1^j-y_2^j=h_j\quad (j=1,2,3) \] has more than $P^\varepsilon$ integral solutions with $1\le x_i,y_i\le P$, then there exist natural numbers $a$ and $b$ with $h_j=a^j-b^j$ $(j=1,2,3)$. This example illustrates the theme that, either the Diophantine system has a paucity of integral solutions, or else the coefficient tuple $\mathbf h$ is highly structured. We examine related paucity problems as well as some consequences for problems involving more variables. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_02911 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Structure and paucity in affine diagonal systems, I Brandes, Julia Wooley, Trevor D. Number Theory 11D45, 11D72, 11D41 Let $\varepsilon>0$ and $\mathbf h\in \mathbb Z^3$. We show that whenever $P$ is large and the system \[ x_1^j+x_2^j-y_1^j-y_2^j=h_j\quad (j=1,2,3) \] has more than $P^\varepsilon$ integral solutions with $1\le x_i,y_i\le P$, then there exist natural numbers $a$ and $b$ with $h_j=a^j-b^j$ $(j=1,2,3)$. This example illustrates the theme that, either the Diophantine system has a paucity of integral solutions, or else the coefficient tuple $\mathbf h$ is highly structured. We examine related paucity problems as well as some consequences for problems involving more variables. |
| title | Structure and paucity in affine diagonal systems, I |
| topic | Number Theory 11D45, 11D72, 11D41 |
| url | https://arxiv.org/abs/2602.02911 |