On the p-adic Waring's problem
Copyright policy )On the p-adic Waring's problem
For any ring \(R\) let \(g_R(n)\) be the least number of \(n\)th powers of elements of \(R\) needed to represent every element of \(R\). In this paper \(R= W(k)\), the unique complete unramified extension of \(\mathbb{Z}_p\) with residue field \(k\), where \(k\) is a given algebraic extension of \(\mathbb{F}_p\). A sample result is the following. If \(k\) is algebraically closed, then \(g_{W(k)}(n)\leq (t+1)^2+1\), where \(t=\nu_p(n)\). The paper is related to that of \textit{J. D. Bovey} [Acta. Arith. 29, 343-351 (1976; Zbl 0336.10014)], and indeed some of Bovey's results are improved.
\(p\)-adic numbers, unramified extension, Waring's problem, Waring's problem and variants, Other analytic theory, Algebraic number theory: local fields
\(p\)-adic numbers, unramified extension, Waring's problem, Waring's problem and variants, Other analytic theory, Algebraic number theory: local fields
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