  We derive the first exact, rigorous but practical, globally valid remainder
terms for asymptotic expansions about saddles and contour endpoints of
arbitrary order degeneracy derived from the method of steepest descents. The
exact remainder terms lead naturally to sharper novel asymptotic bounds for
truncated expansions that are a significant improvement over the previous best
existing bounds for quadratic saddles derived two decades ago. We also develop
a comprehensive hyperasymptotic theory, whereby the remainder terms are
iteratively re-expanded about adjacent saddle points to achieve
better-than-exponential accuracy. By necessity of the degeneracy, the form of
the hyperasymptotic expansions are more complicated than in the case of
quadratic endpoints and saddles, and require generalisations of the
hyperterminants derived in those cases. However we provide efficient methods to
evaluate them, and we remove all possible ambiguities in their definition. We
illustrate this approach for three different examples, providing all the
necessary information for the practical implementation of the method.
