On certain diagonal equations over finite fields

Let @a,@b@?F"q"^"t^* and let N"t(@a,@b) denote the number of solutions (x,y)@?F"q"^"t^*xF"q"^"t^* of the equation x^q^-^1+@ay^q^-^1=@b. Recently, Moisio determined N"2(@a,@b) and evaluated N"3(@a,@b) in terms of the number of rational points on a projective cubic curve over F"q. We show that N"t(@a,@b) can be expressed in terms of the number of monic irreducible polynomials f@?F"q[x] of degree r such that f(0)=a and f(1)=b, where r|t and a,b@?F"q^* are related to @a,@b. 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(@a,@b) can be expressed in terms of the number of monic irreducible cubic polynomials over F"q with certain prescribed trace and norm.

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