A remark on the number field analogue of Waring's constant g(k). Issue 11 (22nd March 2018)

• Record Type: Journal Article
• Authors: Pollack, Paul
• Is Part Of: Mathematische Nachrichten. Volume 291:Issue 11/12(2018)
• Page Start: 1893
• Page End: 1898
• Publication Date: 2018-03-22
• Subjects: Pythagoras number -- Waring's constant -- Waring's problem -- Primary: 11P05; Secondary: 11R04 -- 11R47; Mathematics -- Periodicals; 510.5
• DOI: 10.1002/mana.201700320
• Languages: English
• ISSNs: 0025-584X

Abstract

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 k th powers of elements of O ≥ 0 is the sum of g such k th 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, sup K g ( k, K ) = ∞ .