Estimates on the number of rational solutions of Markoff-Hurwitz equations over finite fields
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866908502240264192 |
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| author | Abdon, Miriam de Oliveira, Daniela Alves Capaverde, Juliane Pérez, Mariana Privitelli, Melina |
| author_facet | Abdon, Miriam de Oliveira, Daniela Alves Capaverde, Juliane Pérez, Mariana Privitelli, Melina |
| contents | Let $N$ denote the number of solutions to the generalized Markoff-Hurwitz-type equation \[(a_1X_1^m+\cdots + a_nX_n^m+a)^k=bX_1\cdots X_n \] over the finite field $\mathbb{F}_q$, where $m,k$ are positive integers, and $a,b,a_i\in \mathbb{F}_q^*$ for $i=1,\dots, n$, with $k,m\ge 2$ and $n\ge 3$. Using techniques from algebraic geometry, we provide an estimate for $N$ and establish conditions under which the equation admits solutions where all $X_i$ are nonzero. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_18191 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Estimates on the number of rational solutions of Markoff-Hurwitz equations over finite fields Abdon, Miriam de Oliveira, Daniela Alves Capaverde, Juliane Pérez, Mariana Privitelli, Melina Number Theory Algebraic Geometry Combinatorics 11G25, 14G05, 14G15 Let $N$ denote the number of solutions to the generalized Markoff-Hurwitz-type equation \[(a_1X_1^m+\cdots + a_nX_n^m+a)^k=bX_1\cdots X_n \] over the finite field $\mathbb{F}_q$, where $m,k$ are positive integers, and $a,b,a_i\in \mathbb{F}_q^*$ for $i=1,\dots, n$, with $k,m\ge 2$ and $n\ge 3$. Using techniques from algebraic geometry, we provide an estimate for $N$ and establish conditions under which the equation admits solutions where all $X_i$ are nonzero. |
| title | Estimates on the number of rational solutions of Markoff-Hurwitz equations over finite fields |
| topic | Number Theory Algebraic Geometry Combinatorics 11G25, 14G05, 14G15 |
| url | https://arxiv.org/abs/2508.18191 |