  We prove that if a homogeneous, continuously differentiable vector field is
asymptotically stable, then it admits a Lyapunov function which is the ratio of
two polynomials (i.e., a rational function). We further show that when the
vector field is polynomial, the Lyapunov inequalities on both the rational
function and its derivative have sum of squares certificates and hence such a
Lyapunov function can always be found by semidefinite programming. This
generalizes the classical fact that an asymptotically stable linear system
admits a quadratic Lyapunov function which satisfies a certain linear matrix
inequality. In addition to homogeneous vector fields, the result can be useful
for showing local asymptotic stability of non-homogeneous systems by proving
asymptotic stability of their lowest order homogeneous component.
  This paper also includes some negative results: We show that (i) in absence
of homogeneity, globally asymptotically stable polynomial vector fields may
fail to admit a global rational Lyapunov function, and (ii) in presence of
homogeneity, the degree of the numerator of a rational Lyapunov function may
need to be arbitrarily high (even for vector fields of fixed degree and
dimension). On the other hand, we also give a family of homogeneous polynomial
vector fields that admit a low-degree rational Lyapunov function but
necessitate polynomial Lyapunov functions of arbitrarily high degree. This
shows the potential benefits of working with rational Lyapunov functions,
particularly as the ones whose existence we guarantee have structured
denominators and are not more expensive to search for than polynomial ones.
