  We present an example of a strictly positive polynomial with rational
coefficients that can be decomposed as a sum of squares of polynomials over
$\R$ but not over $\Q$. This answers an open question by C. Scheiderer posed as
the second question in Section 5.1 of the paper Sums of squares of polynomials
with rational coefficients (2012). We verify that the example we construct
defines a nonsingular projective hypersurface, giving also a positive answer to
the third question posed in that section.
