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In a seminal work of A. N. Varchenko, the behavior at infinity of oscillatoryintegrals with real analytic phase is precisely investigated by using thetheory of toric varieties based on the geometry of the Newton polyhedron of thephase. The purpose of this paper is to generalize his results to the case thatthe phase is contained in a certain class of $C^{\infty}$ functions. The key inour analysis is a toric resolution of singularities in the above class of$C^{\infty}$ functions. The properties of poles of local zeta functions, whichare closely related to the behavior of oscillatory integrals, are also studiedunder the associated situation.

Toric resolution of singularities in a certain class of $C^{\infty}$ functions and asymptotic analysis of oscillatory integrals