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

2007/04/21 by Hillar, Christopher J.
#Commutative Algebra (math.AC) #FOS: Mathematics #Optimization and Control (math.OC) #Rings and Algebras (math.RA)

paper · doi:10.48550/arxiv.0704.2824

Abstract

Let K be a totally real number field with Galois closure L. We prove that if f ∈ \mathbb Q[x₁,...,x_n] is a sum of m squares in K[x₁,...,x_n], then f is a sum of 4m ⋅ 2^{[L: \mathbb Q]+1} [L: \mathbb Q] +1 \choose 2 squares in \mathbb Q[x₁,...,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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