Detecting asymptotic non-regular values by polar curves

TL;DR

This paper introduces a method using affine polar curves to identify Malgrange non-regular values of polynomial functions, including a new concept called super-polar curve, with an effective detection algorithm.

Contribution

It presents a novel approach to detect Malgrange non-regular values via affine polar curves and introduces the super-polar curve for comprehensive identification.

Findings

All non-trivial Malgrange non-regular values are indicated by a single super-polar curve

Provides an effective algorithm for detecting these values

Enhances understanding of polynomial function singularities

Abstract

We locate the Malgrange non-regular values of a given polynomial function $f:\bC^n \to \bC$ by using a series of affine polar curves. We moreover show that all non-trivial Malgrange non-regular values of $f$ are indicated by a single "super-polar curve" which we introduce here, providing also an effective algorithm of detection.