On certain diagonal equations over finite fields

Abstract

Let α , β ∈ F q t ∗ and let N t ( α , β ) denote the number of solutions ( x , y ) ∈ F q t ∗ × F q t ∗ of the equation x q − 1 + α y q − 1 = β . Recently, Moisio determined N 2 ( α , β ) and evaluated N 3 ( α , β ) in terms of the number of rational points on a projective cubic curve over F q . We show that N t ( α , β ) 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 α , β . 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 ( α , β ) can be expressed in terms of the number of monic irreducible cubic polynomials over F q with certain prescribed trace and norm.