A remark on a polynomial mapping $F: \C^n \to \C^{n}$

TL;DR

This paper investigates the topological properties of singular varieties associated with polynomial mappings from complex n-space to itself, establishing non-trivial homology for generic non-proper mappings and providing explicit stratifications for computation.

Contribution

It extends previous results by proving non-trivial homology for a broader class of polynomial mappings and offers explicit stratifications for intersection homology calculations.

Findings

Non-trivial 2-dimensional homology for generic non-proper mappings

Extension of results to higher dimensions ($n o n$)

Explicit stratification for intersection homology computation

Abstract

In \cite{Valette}, Guillaume and Anna Valette associate singular varieties $V_F$ to a polynomial mapping $F: \C^n \to \C^n$. In the case $F: \C^2 \to \C^2$, if the set $K_0(F)$ of critical values of $F$ is empty, then $F$ is not proper if and only if the 2-dimensional homology or intersection homology (with any perversity) of $V_F$ are not trivial. In \cite{ThuyValette}, the results of \cite{Valette} are generalized in the case $F: \C^n \to \C^n$ where $n \geq 3$, with an additional condition. In this paper, we prove that if $F: \C^2 \to \C^2$ is a non-proper {\it generic dominant} polynomial mapping, then the 2-dimensional homology and intersection homology (with any perversity) of $V_F$ are not trivial. We prove that this result is true also for a non-proper {\it generic dominant} polynomial mapping $F: \C^n \to \C^n$ ($\, n \geq 3$), with the same additional condition than in \cite{ThuyValette}. In order to compute the intersection homology of the variety $V_F$, we provide an explicit Thom-Mather stratification of the set $K_0(F) \cup S_F$.