On Waring's Problem in Number Fields

Let K be an algebraic number field of degree n over the rationals, and denote by Jk the subring of K generated by the kth powers of the integers of K. Then GK(k) is defined to be the smallest s⩾1 such that, for all totally positive integers v∈Jk of sufficiently large norm, the Diophantine equation v = λ 1 k + … + λ s k (1.1) is soluble in totally non-negative integers λi of K satisfying N(λi)≪N(v)1/k (1⩽i⩽s). (1.2) In (1.2) and throughout this paper, all implicit constants are assumed to depend only on K, k, and s. The notation GK(k) generalizes the familiar symbol G(k) used in Waring's problem, since we have GQ(k) = G(k). By extending the Hardy–Littlewood circle method to number fields, Siegel [8, 9] initiated a line of research (see [1–4, 11]) which generalized existing methods for treating G(k). This typically led to upper bounds for GK(k) of approximate strength nB(k), where B(k) was the best contemporary upper bound for G(k). For example, Eda [2] gave an extension of Vinogradov's proof (see [13] or [15]) that G(k)⩽(2+o(1))k log k. The present paper will eliminate the need for lengthy generalizations as such, by introducing a new and considerably shorter approach to the problem. Our main result is the following theorem.