International Congress of Mathematicians

208 T. D. WooleyWaring's problem as the basic example. The discussion here illustrates well theissues involved in the analysis of systems of diagonal equations over arbitrary algebraicextensions of Q, and motivates that associated with more general systems ofhomogeneous equations (see [1] and [14]).Let s and k be natural numbers with s > k > 2, and consider an integer nsufficiently large in terms of s and k. The circle method employs Fourier analysisin order to obtain asymptotic information concerning the number, R(n) = R s^(n),of integral solutions of the equation x k + • • • + x k = n. Write F = n x l k and definethe exponential sum f(a) = f(a; P) byf(a) = Y, • 0. Finally, we use the convention that whenever e occurs in a formula, then it isasserted that the statement holds for each fixed positive number e.