Bounds for Waringʼs number modPmin number fields

Abstract

Let P be a prime ideal in the ring of integers R of a number field F , with P ∩ Z = p Z , and assume that P has degree of inertia f and ramification index e < p − 1 . Let γ ( k , P m ) = γ m be the smallest positive integer s such that every integer that is expressible as a sum of k -th powers is a sum of s k -th powers ( mod P m ) . We prove that γ m ⩽ ( γ 1 + 1 2 ) k log ( 2 γ 1 + 1 ) / log p + 1 2 . We also use known bounds for γ 1 to obtain bounds for γ m for any positive integer m .