Generalizations of the Markoff–Hurwitz equations over finite fields

Let NqNq be the number of solutions of the equationa1x12+⋯+anxn2=bx1⋯xn over the finite field Fq=FpsFq=Fps. L. Carlitz found formulas for NqNq for n=3n=3 and for n=4n=4 and q≡3(mod4). He also expressed NqNq for n=4n=4 and q≡1(mod4) in terms of Jacobsthal sums. In an earlier paper, we found formulas for NqNq when d=gcd(n−2,(q−1)/2)=1d=gcd(n−2,(q−1)/2)=1 or 2. In this paper, we find formulas for NqNq when d=4d=4 and p≢7(mod8); and when there exists an l   such that pl≡−1(mod2d).