A remark on the number field analogue of Waring's constant g(k). Issue 11 (22nd March 2018) - Pollack, Paul
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 ) = ∞ .