Geodesic
Sums of squares of polynomials with rational coefficients
Journal of the European Mathematical Society, Tome 18 (2016) no. 7, pp. 1495-1513

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We construct families of explicit (homogeneous) polynomials f over Q that are sums of squares of polynomials over R, but not over Q. Whether or not such examples exist was an open question originally raised by Sturmfels. In the case of ternary quartics we prove that our construction yields all possible examples. We also study representations of the f we construct as sums of squares of rational functions over Q, proving lower bounds for the possible degrees of denominators. For deg(f)=4, or for ternary sextics, we obtain explicit such representations with the minimum degree of the denominators.
DOI : 10.4171/jems/620
Classification : 14-XX, 11-XX, 90-XX
Keywords: Sums of squares, rational coefficients, Hilbert's 17th problem, real plane quartics, exact positivity certificates, semidefinite programming 
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     author = {Claus Scheiderer},
     title = {Sums of squares of polynomials with rational coefficients},
     journal = {Journal of the European Mathematical Society},
     pages = {1495--1513},
     publisher = {mathdoc},
     volume = {18},
     number = {7},
     year = {2016},
     doi = {10.4171/jems/620},
     url = {http://geodesic.mathdoc.fr/articles/10.4171/jems/620/}
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Claus Scheiderer. Sums of squares of polynomials with rational coefficients. Journal of the European Mathematical Society, Tome 18 (2016) no. 7, pp. 1495-1513. doi: 10.4171/jems/620

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