On Waring's Problem in Number Fields

Let K be an algebraic number field of degree n over the rationals, and denote by J k the subring of K generated by the k th powers of the integers of K. Then G K ( k ) is defined to be the smallest s ⩾1 such that, for all totally positive integers v ∈ J k of sufficiently large norm, the Diophantine equation 1.1 v = λ 1 k + … + λ s kis soluble in totally non‐negative integers λ i of K satisfying N(λ i )≪ 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 G K ( 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 ( 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.