On adapted coordinate systems

Abstract

Abstract. The notion of an adapted coordinate system for a given realanalytic function, introduced by V. I. Arnol'd, plays an important role, for instance, 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 analytic functions without multiple components. Varchenko's proof is based on a two-dimensional resolution of singularities result.In this article, we present a more elementary approach to these results, which is based on the Puiseux series expansion of roots of the given function. This approach is inspired by the work of D. H. Phong and E. M. Stein on the Newton polyhedron and oscillatory integral operators. It applies to arbitrary real-analytic functions, and even to arbitrary smooth functions of finite type. In particular, we show that Varchenko's conditions are in fact necessary and sufficient for the adaptedness of a given coordinate system and that adapted coordinates always exist in two dimensions, even in the smooth, finite type setting. For analytic functions, a construction of adapted coordinates by means of Puiseux series expansions of roots has already been carried out in work by

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“…A given coordinate system x is said to be adapted to φ if h(φ) = d x . In [12] we proved that one can always find an adapted local coordinate system in two dimensions, thus generalizing the fundamental work by Varchenko [21] who worked in the setting of real-analytic functions φ. For real analytic functions φ, an alternative proof of Varchenko's result, based on Puiseux series expansions of roots and a clustering of roots has been given by D.H. Phong, E.M. Stein and J. Sturm in [17].…”

“…We also recall from [12] that the homogeneous distance of a κ-homogeneous polynomial P (such as P = φ pr ) is given by d h (P ) := 1/(κ 1 + κ 2 ) = 1/|κ|, and that h(P ) = max{m(P ), d h (P )}.…”

“…According to [12], Corollary 4.3 and Corollary 2.3, the coordinates x are adapted to φ if and only if one of the following conditions is satisfied:…”

“…For real analytic functions φ, an alternative proof of Varchenko's result, based on Puiseux series expansions of roots and a clustering of roots has been given by D.H. Phong, E.M. Stein and J. Sturm in [17]. Our proof in the smooth case in [12] makes use of ideas from both approaches.…”

“…We shall build in this article on the results and techniques developed in [12] and [11], which will be our main references, also for cross-references to earlier and related work. Let us first recall some basic notions from [12], which essentially go back to A.N.…”