# Sums of squares over totally real fields are rational sums of squares - Mathematics Commutative Algebra

Abstract: Let $K$ be a totally real number field with Galois closure $L$. We prove thatif $f \in \mathbb Qx 1, .,x n$ is a sum of $m$ squares in $Kx 1, .,x n$,then $f$ is a sum of \4m \cdot 2^{L: \mathbb Q+1} {L: \mathbb Q +1 \choose2}\ squares in $\mathbb Qx 1, .,x n$. Moreover, our argument isconstructive and generalizes to the case of commutative $K$-algebras. Thisresult gives a partial resolution to a question of Sturmfels on the algebraicdegree of certain semidefinite programing problems.

Author: Christopher J. Hillar

Source: https://arxiv.org/