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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

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


Author: Christopher J. Hillar
Journal: Proc. Amer. Math. Soc. 137 (2009), 921-930
MSC (2000): Primary 12Y05, 12F10, 11E25, 13B24
Published electronically: September 25, 2008
MathSciNet review: 2457431
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Abstract: Let $ K$ be a totally real number field with Galois closure $ L$. We prove that if $ f \in \mathbb{Q}[x_1,\ldots,x_n]$ is a sum of $ m$ squares in $ K[x_1,\ldots,x_n]$, then $ f$ is a sum of

$\displaystyle 4m \cdot 2^{[L: \mathbb{Q}]+1} {[L: \mathbb{Q}] +1 \choose 2}$

squares in $ \mathbb{Q}[x_1,\ldots,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 programming problems.


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Additional Information

Christopher J. Hillar
Affiliation: Department of Mathematics, Texas A&M University, College Station, Texas 77843
Email: chillar@math.tamu.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-08-09641-X
PII: S 0002-9939(08)09641-X
Keywords: Rational sum of squares, semidefinite programming, totally real number field
Received by editor(s): June 11, 2007
Received by editor(s) in revised form: March 31, 2008
Published electronically: September 25, 2008
Additional Notes: The author was supported under a National Science Foundation Postdoctoral Research Fellowship.
Communicated by: Bernd Ulrich
Article copyright: © Copyright 2008 American Mathematical Society