关于自适应坐标系
On adapted coordinate systems
摘要：
自适应坐标系的概念是由V.I.Arnol'd提出的，它在研究振荡积分的渐近展开中起着重要的作用。在二维空间中，A.N.Varchenko给出了给定坐标系自适应的充分条件，并证明了一类无重分量的解析函数的自适应坐标系的存在性。瓦尔琴科的证明是基于Hironaka定理的奇点的分解。在本文中，我们提出了一个新的，基本的和具体的方法，这是基于给定的函数的根的Puixix级数展开。我们的方法适用于任意实解析函数，甚至推广到任意有限型的光滑函数。此外，通过避开Hironaka定理，我们可以给出给定的坐标系在光滑的有限型环境下自适应的充要条件.
英文摘要：
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.
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