Bifurcation values of polynomial functions and perverse sheaves

TL;DR

This paper uses perverse sheaves and vanishing cycles to identify bifurcation values of polynomial functions, providing a new method to compute Euler characteristic jumps and confirming a conjecture in several cases.

Contribution

It introduces a novel approach combining perverse sheaves with Euler characteristic computations to analyze bifurcation values, confirming a conjecture in many instances.

Findings

Characterization of bifurcation values via perverse sheaves

A method to compute Euler characteristic jumps of fibers

Confirmation of Némethi-Zaharia's conjecture in multiple cases

Abstract

We characterize bifurcation values of polynomial functions by using the theory of perverse sheaves and their vanishing cycles. In particular, by introducing a method to compute the jumps of the Euler characteristics with compact support of their fibers, we confirm the conjecture of N\'emethi-Zaharia in many cases.