Sums of squares over totally real fields are rational sums of squares

Details Sums of squares over totally real fields are rational sums of squares Sums of squares over totally real fields are rational sums of squares

2007-04-21

arxiv.org/abs/0704.2824v4

Mathematics

Commutative Algebra

10 pages, final version to appear in Proceedings of the AMS

Scientific paper

Let $K$ be a totally real number field with Galois closure $L$. We prove that if $f \in \mathbb Q[x_1,...,x_n]$ is a sum of $m$ squares in $K[x_1,...,x_n]$, then $f$ is a sum of \[4m \cdot 2^{[L: \mathbb Q]+1} {[L: \mathbb Q] +1 \choose 2}\] squares in $\mathbb Q[x_1,...,x_n]$. Moreover, our argument is constructive and generalizes to the case of commutative $K$-algebras. This result gives a partial resolution to a question of Sturmfels on the algebraic degree of certain semidefinite programing problems.

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