On certain diagonal equations over finite fields

Let alpha, beta is an element of F*(qt) and let N-t(alpha, beta) denote the number of solutions (x, y) is an element of F*(qt) x F*(qt) of the equation x(q-1) + alpha y(q-1) = beta. Recently, Moisio determined N-2(alpha, beta) and evaluated N-3(alpha, beta) in terms of the number of rational points on a projective cubic curve over F-q. We show that N-t(alpha, beta) can be expressed in terms of the number of monic irreducible polynomials f is an element of F-q[x] of degree r such that f(0) = a and f(1) = b, where r vertical bar t and a, b is an element of F*(q) are related to alpha, beta. Let I-r(a, b) denote the number of such polynomials. We prove that I-r(a, b) > 0 when r >= 3. We also show that N-3(alpha, beta) can be expressed in terms of the number of monic irreducible Cubic polynomials over F-q with certain prescribed trace and norm. (C) 2009 Elsevier Inc. All rights reserved.