  We consider the question of when it is possible to force a degenerate scalar
oscillatory integral to decay as fast as a nondegenerate one by restricting the
support to the region where the Hessian determinant of the phase is bounded
below. We show in two dimensions that the desired outcome is not always
possible, but does occur for a broad class of phases which may be described in
terms of the Newton polygon. The estimates obtained are uniform with respect to
linear perturbation of the phase and uniform in the cutoff value of the Hessian
determinant. In the course of the proof, we investigate a
geometrically-invariant approach to making uniform estimates of qualitatively
nondegenerate oscillatory integrals. The approach illuminates a previously
unknown, fundamental relationship between the asymptotics of oscillatory
integrals and the Schr\"odinger equation.
