Tame Functions with strongly isolated singularities at infinity: a tame version of a Parusinski's Theorem

TL;DR

This paper establishes a sufficient condition based on total absolute curvature for the local triviality of definable functions with strongly isolated singularities at infinity, extending Parusinski's Theorem to a tame setting.

Contribution

It provides a tame version of Parusinski's Theorem for complex polynomials with isolated singularities at infinity, using curvature conditions.

Findings

Condition on total absolute curvature ensures local triviality.

Extends Parusinski's Theorem to tame functions.

Applicable to definable functions with isolated singularities at infinity.

Abstract

Let f be a definable function, enough differentiable. Under the condition of having strongly isolated singularities at infinity at a regular value c we give a sufficient condition expressed in terms of the total absolute curvature function to ensure the local triviality of the function f over a neighbourhood of c and doing so providing the tame version of Parusinski's Theorem on complex polynomials with isolated singularities at infinity.