On adapted coordinate systems

TL;DR

This paper introduces a new, elementary approach to understanding adapted coordinate systems for analyzing oscillatory integrals, extending results to smooth functions and providing necessary and sufficient conditions without relying on complex algebraic geometry.

Contribution

It presents a novel, concrete method based on Puiseux series for determining adapted coordinate systems, applicable to real analytic and smooth functions, and removes the need for Hironaka's theorem.

Findings

Provides necessary and sufficient conditions for adaptedness in smooth, finite type functions.

Extends the concept of adapted coordinate systems beyond analytic functions.

Offers an elementary alternative to existing algebraic geometric proofs.

Abstract

The notion of an adapted coordinate system, introduced by V.I.Arnol'd, plays an important role in the study of asymptotic expansions of oscillatory integrals. In two dimensions, A.N.Varchenko gave sufficient conditions for the adaptness of a given coordinate system and proved the existence of an adapted coordinate system for a class of analytic functions without multiple components. Varchenko's proof is based on Hironaka's theorem on the resolution of singularities. In this article, we present a new, elementary and concrete approach to these results, which is based on the Puiseux series expansion of roots of the given function. Our method applies to arbitrary real analytic functions, and even extends to arbitrary smooth functions of finite type. Moreover, by avoiding Hironaka's theorem, we can give necessary and sufficient conditions for the adaptedness of a given coordinate system in the smooth, finite type setting.