  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.
