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 Waring's problem that is to find the minimum number of variables such that the equation x(1)(d) + ... + x(n)(d) = beta has solution for any natural number. In this note we consider a generalization of Waring's problem over finite fields: To find the minimum number delta(k, d, p(f)) of variables such that a system x(1)(k) + ... + x(n)(k) =beta(1), x(1)(k) + ... + x(n)(k) =beta(2) has solution over F-pf for any (beta(1) ,beta(2)) epsilon F-pf(2). We prove that , for p > 3. delta( 1, p(i) +1, p(f)) = 3 if and only if f not equal 2i. We also give an example that proves that, for p = 3, delta(1, 3(i) + 1, 3(f)) >= 4. (c) Elsevier Inc. All rights reserved.