Sums of squares in $S$-integers (2005.06566v2)

Abstract: In totally real number fields, we characterize the rings of $S$-integers (obtained by inverting a rational integer $m$) such that all their totally positive elements are represented as a sum of squares. We further obtain partial answers to the question: when are all the totally positive algebraic integers that are divisible by $m$ represented as a sum of squares?