Topologically principal part of analytic functions

Yoshinaga, Etsuo

doi:10.1090/S0002-9947-1989-0930085-5

Topologically principal part of analytic functions
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by Etsuo Yoshinaga

Trans. Amer. Math. Soc. 314 (1989), 803-814

DOI: https://doi.org/10.1090/S0002-9947-1989-0930085-5

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Abstract:

The problem of ${C^0}$-sufficiency of jets is one of the most interesting problems in the theory of functions or singularities. Roughly speaking, it is the problem of determining a topologically principal part of the Taylor expansion of a given function $f(x)$ at the origin of Euclidean space. Here, the topologically principal part should satisfy the properties that it is as small as possible a part of the Taylor expansion of $f(x)$ and that the local topological type of $f(x)$ at the origin is determined by it. If a function $f(x)$ is an isolated singularity at the origin or has a nondegenerate Newton principal part (see (1.2)), then we know some answers to this problem (see (1.1), (1.3)). The purpose of this paper is to give some results for this problem for any analytic function. The main results are formulated in (1.5), (1.6), and (1.7).

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Newton polyhedra and estimation of oscillating integrals

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