ON WARING'S PROBLEM IN NUMBER FIELDS

Abstract

Let [ ] be an algebraic number field of degree n over the rationals, and denote by J_k the subring of [ ] generated by the kth powers of the integers of [ ]. Then G_{[ ]}(k) is defined to be the smallest s[ges ]1 such that, for all totally positive integers v∈J_k of sufficiently large norm, the Diophantine equation

formula here

is soluble in totally non-negative integers λ_i of [ ] satisfying

formula here

In (1.2) and throughout this paper, all implicit constants are assumed to depend only on [ ], k, and s. The notation G_{[ ]}(k) generalizes the familiar symbol G(k) used in Waring's problem, since we have G_ℚ(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 G_{[ ]}(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)[les ](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.