$E$ Type Singularities for Sufficiently Smooth Functions

TL;DR

This paper characterizes $E$-type singularities in sufficiently smooth functions, providing invariant conditions and normal forms for types $E_6$, $E_7$, and $E_8$, with applications to oscillatory integrals.

Contribution

It introduces invariant criteria for $E$-type singularities in smooth functions and demonstrates reduction to normal forms using the Implicit Function Theorem.

Findings

Invariant conditions for $E_k$ singularities ($6 extle k extle 8)

Normal form reduction for smooth functions

Application to oscillatory integrals

Abstract

In this paper, we will consider $E$-type singularities which are Arnol'd type. We provide invariant conditions for a sufficiently smooth functions to have singularities of type $E_k (6\le k\le 8)$. We show the functions can be reduced to $E_k, k=6, 7, 8$ type normal form under some certain conditions. Moreover, we show that result on normal form for sufficiently smooth functions can be showed by the use of Implicit Function Theorem. The results can be utilized to investigate oscillatory integrals with sufficiently smooth phase function.