  For dispersive Hamiltonian partial differential equations of order 2N+1, N
integer, there are two criteria to analyse to examine the stability of
small-amplitude, periodic travelling wave solutions to high-frequency
perturbations. The first necessary condition for instability is given via the
dispersion relation. The second criterion for instability is the signature of
the eigenvalues of the spectral stability problem given by the sign of the
Hamiltonian. In this work, we show how to combine these two conditions for
instability into a polynomial of degree N. If the polynomial contains no real
roots, then the travelling wave solutions are stable. We present the method for
deriving the polynomial and analyse its roots using Sturm's theory via an
example.
