On Waring's Problem in Number Fields |
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Authors: | Davidson Morley |
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Institution: | Department of Mathematics and Computer Science, Kent State University Kent, OH 44242, USA |
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Abstract: | 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 powersof the integers of K. Then GK(k) is defined to be the smallests 1 such that, for all totally positive integers v Jk of sufficientlylarge norm, the Diophantine equation
(1.1) is soluble in totally non-negative integers i of K satisfying N( i)<<N(v)1/k (1 i s). (1.2) In (1.2) and throughout this paper, all implicit constants areassumed to depend only on K, k, and s. The notation GK(k) generalizesthe familiar symbol G(k) used in Waring's problem, since wehave GQ(k) = G(k). By extending the HardyLittlewood circle method to numberfields, Siegel 8, 9] initiated a line of research (see 14,11]) which generalized existing methods for treating G(k). Thistypically led to upper bounds for GK(k) of approximate strengthnB(k), where B(k) was the best contemporary upper bound forG(k). For example, Eda 2] gave an extension of Vinogradov'sproof (see 13] or 15]) that G(k) (2+o(1))k log k. The presentpaper will eliminate the need for lengthy generalizations assuch, by introducing a new and considerably shorter approachto the problem. Our main result is the following theorem. |
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