Large Improvements in Waring's Problem

Subconvexity for additive equations: Pairs of undenary cubic forms

“…Such is the case, for example, when these polynomials are suitably nonsingular forms in a number of variables exceeding (d−1)2 d−1 , as a consequence of the work of Birch [2], and also when these polynomials are diagonal forms of degree d in d 2 variables (see [33,34]). In the latter case, moreover, if one restricts the variables to be smooth then one can reduce the number of variables required to 1 2 d(log d + log log d + O (1)) (see the methods of [29,30]). It may be worthwhile to be more specific concerning the diagonal examples alluded to above.…”

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“…The principal tool in the proof is the Hardy-Littlewood circle method, incorporating results and techniques of a powerful new iterative method developed by Vaughan and Wooley ([Va3], [Va4], [VW1], [Wo1], [VW2]). …”

“…The supremum of F 1 is estimated by means of either Weyl's inequality for small k or an estimate such as ([Wo1], Theorem 1.4) for large k. The Cauchy-Schwarz inequality may be used to break the integral of |F 2 | into two "mean-square" integrals which may be estimated in an elementary way by consideration of the underlying diophantine equations (cf. Theorem 3 of [Th] and Theorem 6.2 of [Va1]).…”

“…The works of Vaughan and Wooley ([Va3], [Va4], [Wo1], [VW2]) provide, for each pair of positive integers (k, s), an estimate k, s). Note that we may take λ(k, 1) = 1 and λ(k, 2) = 2 + ε.…”

“…The values for 5 k 9 are taken from the appendix of [VW2]. The remaining values are the result of an iteration procedure (on s), applying at each step one of Lemma 2.2 of [Va4] (for s = 5 and some k), Lemma 3.2 of [Wo1] …”