Uniform (m)-condition and Strong Milnor fibrations

TL;DR

This paper investigates the conditions under which real analytic map germs from olds to orms admit Strong Milnor fibrations, linking their existence to transversality conditions of associated analytic varieties.

Contribution

It establishes a connection between Strong Milnor fibrations and transversality conditions of a family of analytic varieties derived from the map germ.

Findings

Characterizes Strong Milnor fibrations via transversality conditions.

Relates the existence of fibrations to geometric properties of associated varieties.

Provides criteria for the existence of Milnor fibrations in real analytic settings.

Abstract

In this paper we study the Milnor fibrations associated to real analytic map germs $\psi:(\mathbb{R}^{m},0) \to (\mathbb{R}^2,0)$ with isolated critical point at $0\in \mathbb{R}^{m}$. The main result relates the existence of called Strong Milnor fibrations with a transversality condition of a convenient family of analytic varieties with isolated critical points at the origin $0\in \mathbb{R}^{m}$, obtained by projecting the map germ $\psi$ in the family $L_{-\theta}$ of all lines through the origin in the plane $\mathbb R^{2}.$