Explicit analytic functions defining the images of wave-front singularities

TL;DR

This paper constructs explicit real-analytic functions that precisely describe the images of wave-front singularities, providing a new framework based on resultant computations for understanding these geometric structures.

Contribution

It introduces a general method for constructing main-analytic functions for wave-front singularities using explicit resultant calculations, with formulas for types A, D, and E.

Findings

Explicit formulas for main-analytic functions of wave-front singularities

A new framework for constructing these functions using resultants

Characterization of images of wave-front singularities in Euclidean space

Abstract

We give explicit real-analytic functions whose zero sets characterize the images of the standard maps of wave-front singularities. Such functions are realizations of the main-analytic sets in the sense of Ishikawa-Koike-Shiota (1984). More concretely, a subset of Euclidean space is called a global main-analytic set if it can be described, up to a set of smaller Hausdorff dimension, as part of the zero set of a single real-analytic function, referred to as its main-analytic function. In this paper, we propose a general framework for constructing main-analytic functions by a method based on explicit resultant computations. In particular, we provide explicit formulas for the main-analytic functions associated with the standard maps of wave-front singularities of types A, D and E.