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arXiv · 2508.18191

Estimates on the number of rational solutions of Markoff-Hurwitz equations over finite fields

Miriam Abdon·Daniela Alves de Oliveira·Juliane Capaverde·Mariana Pérez·Melina Privitelli
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Abstract

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.

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Miriam Abdon, Daniela Alves de Oliveira, Juliane Capaverde, Mariana Pérez, Melina Privitelli. 2025-08-25. Estimates on the number of rational solutions of Markoff-Hurwitz equations over finite fields. https://arxiv.org/abs/2508.18191

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