  We derive some Positivstellensatz\"e for noncommutative rational expressions
from the Positivstellensatz\"e for noncommutative polynomials. Specifically, we
show that if a noncommutative rational expression is positive on a polynomially
convex set, then there is an algebraic certificate witnessing that fact. As in
the case of noncommutative polynomials, our results are nicer when we
additionally assume positivity on a convex set-- that is, we obtain a so-called
"perfect Positivstellensatz" on convex sets.
