On generalized Markoff-Hurwitz-type equations over finite fields
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 is the augmented degree matrix. Under the assumption of \(\gcd\left(\sum_{i=1}^n k_i m/m_i -m, q-1\right) =1\) where \(m=m_1\cdots m_n\), the augmented degree matrix is invertible. This shows that \(N_q\) is independent of the coefficients \(a_i\) and \(b\) and therefore \(N_q = q^{n-1} + (-1)^{n-1}\). The expression of \(N_q\) become very simple in this case.
scientific article; zbMATH DE number 5822859
scientific article