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A remark on the number field analogue of Waring's constant g(k)

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  • Paul Pollack

Abstract

Let K be a number field, and let k be an integer with k≥2. Let O≥0 be the collection of totally nonnegative integers in K (i.e., the totally positive integers together with zero). We let g(k,K) denote the smallest positive integer with the following property: Every element of O≥0 that is a sum of kth powers of elements of O≥0 is the sum of g such kth powers. Work of Siegel in the 1940s shows that g(k,K) is well‐defined for all k and K. In this note, we prove that g(k,K) cannot be bounded by a function of k alone: For each k≥2, supKg(k,K)=∞.

Suggested Citation

  • Paul Pollack, 2018. "A remark on the number field analogue of Waring's constant g(k)," Mathematische Nachrichten, Wiley Blackwell, vol. 291(11-12), pages 1893-1898, August.
    • Handle: RePEc:bla:mathna:v:291:y:2018:i:11-12:p:1893-1898
    DOI: 10.1002/mana.201700320
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