On stable polynomial mappings

Michał Farnik; Zbigniew Jelonek

paper

On stable polynomial mappings

arXiv:2006.09039

Abstract

For given natural numbers $d_{1}, d_{2}$ let $Ω_{2} (d_{1}, d_{2})$ be the set off all polynomial mappings $F = (f, g) : C^{2} \to C^{2}$ such that deg $f \leq d_{1}$ , deg $g \leq d_{2}$ . We say that the mapping $F$ is topologically stable in $Ω_{2} (d_{1}, d_{2})$ if for every small deformation $F_{t} \in Ω_{2} (d_{1}, d_{2})$ the mapping $F_{t}$ is topologically equivalent to the mapping $F$ . The aim of this paper is to characterize the topologically stable mappings in $Ω_{2} (d_{1}, d_{2})$ . In particular we show how to effectively determine a member of $Ω_{2} (d_{1}, d_{2})$ with generic topology.