Suppose we are given a univariate polynomial $p(x)$ with rational coefficients, and are told that $p(x)$ is SOS. It is well-known that every univariate SOS polynomial can be expressed as a sum of two squares. Can we efficiently find a representation of $p(x)$ as $f^2(x) + g^2(x)$, where both f and g have *rational* coefficients? Just to be clear: I want an efficient algorithm which takes as input an SOS polynomial $p(x)$ with rational coefficients and outputs two polynomials $f(x), g(x)$ with rational coefficients such that $p(x) = f^2(x) + g^2(x)$.