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$E$ Type Singularities for Sufficiently Smooth Functions

Ibrokhimbek Akramov, Dildora Ikromova

TL;DR

This work defines invariant criteria for Arnold's $E$-type singularities ($E_6$, $E_7$, $E_8$) of sufficiently smooth functions with a critical point at the origin and proves that, under nondegenerate linear changes of variables, these functions admit simple normal forms related to oscillatory integrals. The authors formalize multiplicity and resultant-based conditions using local algebras and $p_k$-polynomials, and they construct explicit diffeomorphisms to reach canonical normal forms: $\pm y_1^4 + y_2^3$ for $E_6$, $y_2^3 + y_2 y_1^3$ for $E_7$, and $y_1^5 + y_2^3$ for $E_8$. They provide constructive proofs for $C^7$–$C^9$ smoothness levels and discuss implications for invariant analyses of oscillatory integrals and Fourier restriction problems. The results offer practical normal-form reductions that facilitate asymptotic and harmonic-analytic investigations of phase functions near high-order singularities.

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.

$E$ Type Singularities for Sufficiently Smooth Functions

TL;DR

This work defines invariant criteria for Arnold's -type singularities (, , ) of sufficiently smooth functions with a critical point at the origin and proves that, under nondegenerate linear changes of variables, these functions admit simple normal forms related to oscillatory integrals. The authors formalize multiplicity and resultant-based conditions using local algebras and -polynomials, and they construct explicit diffeomorphisms to reach canonical normal forms: for , for , and for . They provide constructive proofs for – smoothness levels and discuss implications for invariant analyses of oscillatory integrals and Fourier restriction problems. The results offer practical normal-form reductions that facilitate asymptotic and harmonic-analytic investigations of phase functions near high-order singularities.

Abstract

In this paper, we will consider -type singularities which are Arnol'd type. We provide invariant conditions for a sufficiently smooth functions to have singularities of type . We show the functions can be reduced to 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.

Paper Structure

This paper contains 5 sections, 8 theorems, 71 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Invariant definition of $E_6$ type singularities
  4. The case of $E_7$ type singularities
  5. Invariant definition of $E_8$ type singularities

Key Result

Lemma 5

Let $\phi$ be a $C^3(U)$ function in a neighborhood $U$ of the origin satisfying Then there exists a neighborhood $V\subset U$ and a $C^1$ function $\psi$ defined in $V$ such that $\psi(0)=\psi'(0)=0$ and where $B$ is a continuous function with $B(0, 0)\neq0$.

Theorems & Definitions (18)

  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Lemma 5
  • proof
  • Corollary 6
  • Theorem 7
  • Remark 8
  • proof
  • ...and 8 more