# Sums of squares of polynomials with rational coefficients

@article{Scheiderer2012SumsOS,
title={Sums of squares of polynomials with rational coefficients},
author={Claus Scheiderer},
journal={arXiv: Algebraic Geometry},
year={2012}
}
We construct families of explicit polynomials f with rational coefficients that are sums of squares of polynomials over the real numbers, but not over the rational numbers. Whether or not such examples exist was an open question originally raised by Sturmfels. We also study representations of f as sums of squares of rational functions with rational coefficients. In the case of ternary quartics, we prove that our counterexamples to Sturmfels' question are the only ones.
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#### References

SHOWING 1-10 OF 25 REFERENCES
SUMS OF SQUARES OVER TOTALLY REAL FIELDS ARE RATIONAL SUMS OF SQUARES
Let K be a totally real number field with Galois closure L. We prove that if f ∈ Q(x 1 ,.., x n ] is a sum of m squares in K[x 1 ,.., x n ], then f is a sum of 4m·2 [L:Q]+1 ( [L:Q+1 2 ) squares inExpand
Exact certification in global polynomial optimization via sums-of-squares of rational functions with rational coefficients
• Mathematics, Computer Science
• J. Symb. Comput.
• 2012
We present a hybrid symbolic-numeric algorithm for certifying a polynomial or rational function with rational coefficients to be non-negative for all real values of the variables by computing aExpand
Tight bounds for rational sums of squares over totally real fields
AbstractLet K be a totally real Galois number field. Hillar proved that if f ∈ ℚ[x1, ..., xn] is a sum of m squares in K[x1, ..., xn], then f is a sum of N(m) squares in ℚ[x1, ..., xn], where N(m) ≤Expand
This chapter presents quadratic and hermitian forms. The chapter also presents a theorem which states that the inverse of a nonsingular linear transformation is a nonsingular linear transformation.Expand
Moments, Positive Polynomials And Their Applications
. From a theoretical viewpoint, the GPM has developments and impact in var-ious area of Mathematics like algebra, Fourier analysis, functional analysis, operator theory, probabilityand statistics, toExpand
Computing Rational Points in Convex Semialgebraic Sets and Sum of Squares Decompositions
• Computer Science, Mathematics
• SIAM J. Optim.
• 2010
An algorithm returning a rational point in $\mathcal{S}$ if and only if $S}\cap\mathbb{Q}\neq\emptyset$ requires bit operations and the coefficients of the outputted polynomials have bit length dominated by $\tau D^{O}(k^3)}$. Expand
Convex Optimization
• Mathematics, Computer Science
• IEEE Transactions on Automatic Control
• 2006
A comprehensive introduction to the subject of convex optimization shows in detail how such problems can be solved numerically with great efficiency. Expand
Handbook on semidefinite, conic and polynomial optimization
• Mathematics
• 2012
Introduction to Semidefinite, Conic and Polynomial Optimization.- The Approach of Moments for Polynomial Equations.- Algebraic Degree in Semidefinite and Polynomial Optimization.- SemidefiniteExpand
Handbook of semidefinite programming : theory, algorithms, and applications
• Mathematics
• 2000
Contributing Authors. List of Figures. List of Tables. Preface. 1. Introduction H. Wolkowicz, et al. Part I: Theory. 2. Convex Analysis on Symmetric Matrices F. Jarre. 3. The Geometry of SemidefiniteExpand
Advances in convex optimization: conic programming
During the last two decades, major developments in convex optimization were focusing on conic programming, primarily, on linear, conic quadratic and semidefinite optimization. Conic programmingExpand