Nonhomogeneous waring equations

On Waring's problem: some consequences of Golubeva's method

“…Our goal in this section and the next is the proof of Theorem 1.2, together with the conclusion of Theorem 1.1 which essentially amounts to a corollary of the former theorem. Here we make use of a method originating in work of Linnik [15], and much enhanced by the level-lowering procedure of Golubeva [7,8]. We seek to establish the solubility of the equation (1.4) when n is large by putting x 3 = A + x 0 and x 4 = A − x 0 , with A ∈ N of size nearly n 1/3 and |x 0 | < A.…”

“…Rather than apply the circle method to establish Theorem 1.1, which as we have noted is limited to situations with β(k) > 2 by the convexity barrier, we instead apply a method of Golubeva involving the theory of ternary quadratic forms [7,8]. In applying this method, we avoid in this paper certain difficulties arising from auxiliary congruences by assuming where necessary the truth of GRH.…”

“…Subject to natural conditions of congruential type, Gauss [6] tackled the case k = (2, 2, 2) with β(k) = 3 2 , and more recently Linnik [16] and Hooley [11] successfully considered k = (2, 2, 3, 3, 3) with β(k) = 2. Also, work of Golubeva [7,8] addresses the mixed exponent k = (2, 2, 3, 3, 4, 16, 4k + 1) with β(k) = 2 − 1 48 + 1 4k+1 . Most recently of all, subject to the validity of the Elliott-Halberstam conjecture together with GRH, work of Friedlander and the author [4] resolves the Waring problem corresponding to the exponent tuple k = (2, 2, 4, 4, 4, k), with β(k) = 2 − 1 4 + 1 k .…”

On Waring's Problem: Two Squares and Three Biquadrates

“…On the other hand, Linnik [15] and Hooley [11] investigated sums of two squares and three cubes. Very recently, Golubeva [8,9] has shown that all large integers n are represented as a sum of positive integral powers in the shape…”