On generalized Markoff-Hurwitz-type equations over finite fields
type: Item
instance of: scholarly article
title: On generalized Markoff-Hurwitz-type equations over finite fields
zbMATH Open document ID: 1218.11038
author: Wei Cao
published in: Acta Applicandae Mathematicae
publication date: 2010-12-03T00:00:00Z
review text: Let \(N_q\) denote the number of solutions of generalized Markoff-Hurwitz-type equations of the form \[ a_1 x_1^{m_1} + a_2 x_2^{m_2} + \cdots + a_n x_n^{m_n} = b x_1^{k_1} x_2^{k_2} \dots x_n^{k_n} \] over the finite field \(\mathbb{F}_q\), where \(m_i, k_i\) are positive integers, \(a_i, b \in \mathbb{F}_q^*\) for \(i = 1, \ldots, n\) and \(n\geq 2\). When \(m_1 = m_2 = \cdots = m_n =2\), \(k_1 = k_2 = \cdots = k_n =1\) and \(a_1= a_2= \cdots = a_n =1\), these equations are called Markoff-Hurwitz equations which were first studied by Markoff for the case \(n=3\) and \(b=3\) and then by Hurwitz more generally. \textit{L. Carlitz} [Monatsh. Math. 58, 5--12 (1954; Zbl 0055.26803)] extended Markoff-Hurwitz-type equations by varying \(a_i\)'s and obtained explicit formulas for \(N_q\) in terms of quadratic characters or Jacobsthal sums for \(n=3\) and \(n=4\). When \(k_1= k_2 = \cdots = k_n =1\), \(m_i \mid q-1\), and \(a_i, b\in \mathbb{F}_q^*\), \textit{Yu. N. Baoulina} (Baulina) [Act. Appl. Math. 85, 35--39 (2005; Zbl 1126.11019)] studied \(N_q\) for the generalized Markoff-Hurwitz-type equations under the condition \(d = \gcd\left(\sum_{i=1}^n m/m_i- m, (q-1)/m\right) =1\) with \(m= \mathrm{lcm}(m_1, \dots, m_n)\). In the case that \(a_1=a_2=\cdots =a_n =1\), \textit{Yu. N. Baoulina} [Finite Fields Appl. 13, 887--895 (2007; Zbl 1167.11012)] also obtained the formula for \(N_q\) when \(md > 2\) and there exists an \(\ell\) such that \(md \mid (p^{\ell} +1)\). This paper deals with arbitrary positive integers \(m_i, k_i\) and \(a_i, b \in \mathbb{F}_q^*\) under the condition \(\gcd\left(\sum_{i=1}^n k_i m/m_i - m, q-1\right) =1\) where \(m= m_1\cdots m_n\). By introducing one extra variable \(x_0\) into the multivariate polynomial equation, a square augmented degree matrix is obtained. Using the results of Gauss sums, the problem of finding \(N_q\) is reduced to solve a homogeneous system of \(n+1\) linear equations in \(\mathbb{Z}_{q-1}\), whose coefficient matrix i…
Mathematics Subject Classification ID: 11D79
Mathematics Subject Classification ID: 11D45
Mathematics Subject Classification ID: 11T06
Mathematics Subject Classification ID: 11T24
zbMATH DE Number: 5822859
zbMATH Keywords: finite fields
zbMATH Keywords: Markoff-Hurwitz type equations
zbMATH Keywords: augmented degree matrix
zbMATH Keywords: Gauss sum
reviewed by: Qiang Wang
MaRDI profile type: Publication
full work available at URL: s10440-010-9568-4
OpenAlex ID: W2005039466
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DOI: 10.1007/S10440-010-9568-4
description: scientific article; zbMATH DE number 5822859