Solving sums of squares in global fields

TL;DR

This paper introduces an explicit algorithm for decomposing elements into sums of squares in any global field, advancing the understanding of positive element representations in algebraic number theory.

Contribution

It provides the first general algorithm applicable to all global fields for expressing elements as sums of squares of minimal length.

Findings

Algorithm works for any global field including number fields and function fields.

Decomposition into sums of squares achieved with minimal length.

Enhances methods for positive element representation in algebraic structures.

Abstract

The problem of writing a totally positive element as a sum of squares has a long history in mathematics, going back to Bachet and Lagrange. While for some specific rings (like integers or polynomials over the rationals), there are known methods for decomposing an element into a sum of squares, in general, for many other important rings and fields, the problem is still widely open. In this paper, we present an explicit algorithm for decomposing an element of an arbitrary global field (either a number field or a global function field) into a sum of squares of minimal length.