  We study the problem of decomposing a non-negative polynomial as an exact sum
of squares (SOS) in the case where the associated semidefinite program is
feasible but not strictly feasible (for example if the polynomial has real
zeros). Computing symbolically roots of the original polynomial and applying
facial reduction techniques, we can solve the problem algebraically or restrict
to a subspace where the problem becomes strictly feasible and a numerical
approximation can be rounded to an exact solution.
  As an application, we study the problem of determining when can a rational
polynomial that is a sum of squares of polynomials with real coefficients be
written as sum of squares of polynomials with rational coefficients, and answer
this question for some previously unknown cases. We first prove that if $f$ is
the sum of two squares with coefficients in an algebraic extension of ${\mathbb
Q}$ of odd degree, then it can always be decomposed as a rational SOS. For the
case of more than two polynomials we provide an example of an irreducible
polynomial that is the sum of three squares with coefficients in ${\mathbb
Q}(\sqrt[3]{2})$ that cannot be decomposed as a rational SOS.
