On systems of linear and diagonal equation of degree [formula omitted] over finite fields of characteristic p

Abstract

One of the most important questions in number theory is to find properties on a system of equations that guarantee solutions over a field. A well-known problem is Waring's problem that is to find the minimum number of variables such that the equation x 1 d + ⋯ + x n d = β has solution for any natural number β. In this note we consider a generalization of Waring's problem over finite fields: To find the minimum number δ ( k , d , p f ) of variables such that a system x 1 k + ⋯ + x n k = β 1 , x 1 d + ⋯ + x n d = β 2 has solution over F p f for any ( β 1 , β 2 ) ∈ F p f 2 . We prove that, for p > 3 , δ ( 1 , p i + 1 , p f ) = 3 if and only if f ≠ 2 i . We also give an example that proves that, for p = 3 , δ ( 1 , 3 i + 1 , 3 f ) ⩾ 4 .