Waring’s Problem

Wang Yuan
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Wang Yuan: Academia Sinica, Institute of Mathematics

Chapter Chapter 7 in Diophantine Equations and Inequalities in Algebraic Number Fields, 1991, pp 87-97 from Springer

Abstract: Abstract Waring’s problem in an algebraic number field is to consider the problem of decomposing a totally nonnegative integer ν as a sum of k-th powers of totally nonnegative integers, namely 7.1 $$ v = \lambda _1^k + \cdots + \lambda _s^k, $$ where (λ1,…,λs) ∈ P s . It was pointed out by Siegel that there may exist infinitely many integers in P which are not sums of k-th powers. This led him to consider the ring J k generated by k-th powers of integers instead of P; see Introduction. Suppose that ν ∈ J k ∩P. Let r s (ν) be the number of solutions of the equation (7.1) subject to the condition λ i ∈P(T), 1≤i≤s.

Date: 1991
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DOI: 10.1007/978-3-642-58171-7_7

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