  A theorem of Varchenko gives the order of decay of the leading term of the
asymptotic expansion of a degenerate oscillatory integral with real-analytic
phase in two dimensions. His theorem expresses this order of decay in a simple
geometric way in terms of its Newton polygon once one is in certain coordinate
systems called adapted coordinate systems.
  In this paper, we give explicit formulas that not only provide the order of
decay of the leading term, but also the coefficient of this term. There are
three rather different formulas corresponding to three different types of
Newton polygon. Analogous results for sublevel integrals are proven, as are
analogues for the more general case of smooth phase. The formulas require one
to be in certain "superadapted" coordinates. These are a type of adapted
coordinate system which we show exists for any smooth phase.

Classical Analysis and ODEs