The number of solutions of certain equations over a finite field

Let F be a finite field with q=pf elements, where p is a prime number. Let N(n) be the number of solutions (x1,…,xn) of the triangular equation a1x1d11+a2x1d21x2d22+⋯+anx1dn1⋯xndnn=b in Fqn, where n⩾2,dij⩾0,ai∈Fq* and b∈Fq. In this paper, we obtain an explicit formula for the expression N(n) under the necessary restriction gcd(d11d22…dnn,q-1)=1. We also discuss the general case without the additional restriction by applying Jean Delsarte's method of character sum techniques (Nombre de solutions des équations polynomiales sur un corps fini, Séminaire Bourbaki, Exposé, 39:1C9, March 1951; An English summary available at: http://arxiv.org/PS_cache/math/pdf/0401/0401066.pdf).