International Congress of Mathematicians

212 T. D. Wooleywith 1 < z,w < F, M < m < MR, (ip'(z)ip'(w), m) = 1 and m,Vi £ A(T/M,R)(1 < i < s). The implicit congruence condition ip(z) = ip(w) (mod m k ) maybeanalytically refined to the stronger one z = w (mod m k ), and in this wayoneis led to replace the expression ip(z) — ip(w) by the difference polynomialip\(z;h;m) = mr k (tp(z + hm k ) — fp(zj). Notice that when M > P x l k , one isforced to conclude that z = w, and then the number of solutions of (2.4) is boundedabove by PMRS S (T/M,R) < P 1+t M(T/M) x '. Otherwise, following an applicationof Schwarz's inequality to the associated mean value of exponential sums, onemay recover an equation of the shape (2.3) in which ip(z) is replaced by ip\(z), andT is replaced by T/M, and repeat the process once again. This gives a repeateddifferencing process that hybridises that of Weyl with the ideas of Vinogradov.It is now possible to describe a strategy for bounding a permissible exponentA s+ i in terms of a known permissible exponent A s . We initially take T = Pand ip(z) = z k , and observe that S s+ i(P,R) is bounded above by the number ofsolutions of (2.3). We apply the above efficient differencing process successivelywith appropriate choices for M at each stage, say M = P^1,with 0 <