On systems of linear and diagonal equation of degree pi+1pi+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 Waring's problem that is to find the minimum number of variables such that the equation x1d+⋯+xnd=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,pf)δ(k,d,pf) of variables such that a systemx1k+⋯+xnk=β1,x1d+⋯+xnd=β2 has solution over FpfFpf for any (β1,β2)∈Fpf2. We prove that, for p>3p>3, δ(1,pi+1,pf)=3δ(1,pi+1,pf)=3 if and only if f≠2if≠2i. We also give an example that proves that, for p=3p=3, δ(1,i3+1,f3)⩾4δ(1,3i+1,3f)⩾4.