Generalizations of the Markoff-Hurwitz equations over finite fields (Q2493050)

Let \(\mathbb F_q\) be a finite field with \(q=p^s\) elements and let \(N_q\) be the number of solutions in \(\mathbb F_q\) of \(\sum_{i=1}^n a_ix_i^2=bx_1\cdots x_n\). \textit{L. Carlitz} [Monatsh. Math. 58, 5--12 (1954; Zbl 0055.26803)] gave formulas for \(N_q\) when \(n=3\) and when \(n=4\), \(q\equiv 3\pmod{4}\). In a previous paper [Current trends in number theory. Proceedings of the international conference on number theory, Allahabad, India, November 2000. New Delhi: Hindustan Book Agency, 27--37 (2002; Zbl 1086.11021)], the author found explicit formulas for \(N_q\) when \(d=(n-2, (q-1)/2)=1\) or~\(2\). Here formulas are found when \(d=4\) and \(p\not\equiv 7\pmod{8}\) and when a power of \(p\) is equivalent to \(-1\) modulo \(2d\). The results are obtained by determining certain Gauss sums.