  We deploy numerical semidefinite programming and conversion to exact rational
inequalities to certify that for a positive semidefinite input polynomial or
rational function, any representation as a fraction of sums-of-squares of
polynomials with real coefficients must contain polynomials in the denominator
of degree no less than a given input lower bound. By Artin's solution to
Hilbert's 17th problems, such representations always exist for some denominator
degree. Our certificates of infeasibility are based on the generalization of
Farkas's Lemma to semidefinite programming.
  The literature has many famous examples of impossibility of SOS
representability including Motzkin's, Robinson's, Choi's and Lam's polynomials,
and Reznick's lower degree bounds on uniform denominators, e.g., powers of the
sum-of-squares of each variable. Our work on exact certificates for positive
semidefiniteness allows for non-uniform denominators, which can have lower
degree and are often easier to convert to exact identities. Here we demonstrate
our algorithm by computing certificates of impossibilities for an arbitrary
sum-of-squares denominator of degree 2 and 4 for some symmetric sextics in 4
and 5 variables, respectively. We can also certify impossibility of base
polynomials in the denominator of restricted term structure, for instance as in
Landau's reduction by one less variable.
