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

TL;DR

This paper provides estimates for the number of solutions to generalized Markoff-Hurwitz equations over finite fields, using algebraic geometry techniques to analyze solution counts and conditions for nonzero solutions.

Contribution

It introduces new bounds and conditions for solutions of Markoff-Hurwitz equations over finite fields, extending previous understanding with algebraic geometry methods.

Findings

Derived estimates for the number of solutions over finite fields

Established conditions for solutions with all variables nonzero

Applied algebraic geometry techniques to analyze solution structure

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.