On certain diagonal equations over finite fields

Let and let Nt(α,β) denote the number of solutions of the equation xq−1+αyq−1=β. Recently, Moisio determined N2(α,β) and evaluated N3(α,β) in terms of the number of rational points on a projective cubic curve over Fq. We show that Nt(α,β) can be expressed in terms of the number of monic irreducible polynomials f∈Fq[x] of degree r such that f(0)=a and f(1)=b, where r|t and are related to α,β. Let Ir(a,b) denote the number of such polynomials. We prove that Ir(a,b)>0 when r⩾3. We also show that N3(α,β) can be expressed in terms of the number of monic irreducible cubic polynomials over Fq with certain prescribed trace and norm.