Toric resolution of singularities in a certain class of \(C^{\infty}\) functions and asymptotic analysis of oscillatory integrals (Q2814328)

The authors generalize the results of \textit{A. N. Varchenko} [Functional Anal. Appl. 10, No. 3, 175--196 (1976)], under the same nondegeneracy hypothesis, to the case that the phase is contained in a certain class of \(C^{\infty}\) functions including real analytic functions. This class is denoted by \(\hat{\mathcal{E}}(U)\), where \(U\) is an open neighborhood of the origin in \(\mathbb{R}^n\). Under the nondegeneracy condition, they construct a toric resolution of singularities in the class \(\hat{\mathcal{E}}(U)\).NEWLINENEWLINEThe paper is organized as follows : Introduction, Newton polyhedra and classes \(\hat{\mathcal{E}}[P](U)\) and \(\hat{\mathcal{E}}(U)\), main results, lemmas on polyhedra, remarks on the \(\gamma\)-part, properties of \(\hat{\mathcal{E}}[P](U)\) and \(\hat{\mathcal{E}}(U)\), toric varieties constructed from polyhedra, toric resolution of singularities in the class \(\hat{\mathcal{E}}(U)\), poles of local zeta functions, proofs of the theorems in Section 3, examples.NEWLINENEWLINEOther authors articles directly connected to this topic are [in: Topics in finite or infinite dimensional complex analysis. Proceedings of the 19th international conference on finite or infinite dimensional complex analysis and applications, ICFIDCAA'11 Sendai: Tohoku University Press. 3--12 (2013; Zbl 1298.42019); RIMS Kôkyûroku Bessatsu B40, 31--40 (2013; Zbl 1290.42038)].