Sums of squares over totally real fields are rational sums of squares
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- by Christopher J. Hillar PDF
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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 \[ 4m \cdot 2^{[L: \mathbb Q]+1} {[L: \mathbb Q] +1 \choose 2}\] 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.References
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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
- 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
- © Copyright 2008 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 137 (2009), 921-930
- MSC (2000): Primary 12Y05, 12F10, 11E25, 13B24
- DOI: https://doi.org/10.1090/S0002-9939-08-09641-X
- MathSciNet review: 2457431