Sums of squares of polynomials with rational coefficients

Abstract

We construct families of explicit (homogeneous) polynomials $f$ over $\mathbb{Q}$ that are sums of squares of polynomials over $\mathbb{R}$, but not over $\mathbb{Q}$. 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 $\mathbb{Q}$, 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.