TY - JOUR

T1 - Toric resolution of singularities in a certain class of C∞ functions and asymptotic analysis of oscillatory integrals

AU - Kamimoto, Joe

AU - Nose, Toshihiro

N1 - Funding Information:
The first author was supported by Grant-in-Aid for Scientific Research (C) (No. 22540199, 15K04932), Japan Society for the Promotion of Science. The second author was supported by Grant-in-Aid for Young Scientists (B) (No. 15K17565), Japan Society for the Promotion of Science.

PY - 2016

Y1 - 2016

N2 - In a seminal work of A. N. Varchenko, the behavior at infinity of oscillatory integrals with real analytic phase is precisely investigated by using the theory of toric varieties based on the geometry of the Newton polyhedron of the phase. The purpose of this paper is to generalize his results to the case that the phase is contained in a certain class of C∞ functions. The key in our analysis is a toric resolution of singularities in the above class of C∞ functions. The properties of poles of local zeta functions, which are closely related to the behavior of oscillatory integrals, are also studied under the associated situation.

AB - In a seminal work of A. N. Varchenko, the behavior at infinity of oscillatory integrals with real analytic phase is precisely investigated by using the theory of toric varieties based on the geometry of the Newton polyhedron of the phase. The purpose of this paper is to generalize his results to the case that the phase is contained in a certain class of C∞ functions. The key in our analysis is a toric resolution of singularities in the above class of C∞ functions. The properties of poles of local zeta functions, which are closely related to the behavior of oscillatory integrals, are also studied under the associated situation.

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M3 - Review article

AN - SCOPUS:85047304169

VL - 23

SP - 425

EP - 485

JO - Journal of Mathematical Sciences (Japan)

JF - Journal of Mathematical Sciences (Japan)

SN - 1340-5705

IS - 2

ER -