The A_f condition and relative conormal spaces for functions with non-vanishing derivative

TL;DR

This paper introduces a join construction to better understand the relative conormal space of analytic functions with non-zero derivatives and provides a numerical criterion for Thom's A_f condition.

Contribution

It presents a novel join construction for relative conormal spaces and establishes a numerical criterion for Thom's A_f condition in complex analytic spaces.

Findings

Join construction effectively completes the description of relative conormal spaces.

Numerical criterion for Thom's A_f condition is derived.

Application to functions with non-vanishing derivatives at the origin.

Abstract

We introduce a join construction as a way of completing the description of the relative conormal space of an analytic function on a complex analytic space that has a non-vanishing derivative at the origin. Then we show how to obtain a numerical criterion for Thom's A_f condition.