On systems of linear and diagonal equation of degree pi+1 over finite fields of characteristic p

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 Waringʹs problem that is to find the minimum number of variables such that the equation has solution for any natural number β. In this note we consider a generalization of Waringʹs problem over finite fields: To find the minimum number δ(k,d,pf) of variables such that a system has solution over for any . We prove that, for p>3, δ(1,pi+1,pf)=3 if and only if f≠2i. We also give an example that proves that, for p=3, δ(1,3i+1,3f) 4.