Topological triviality and link-constancy in deformations of inner Khovanskii non-degenerate maps

Abstract

For real and mixed polynomial maps $f=(f^1,\dots,f^p)$ satisfying $f(0)=0$, we introduce the notion of Inner Khovanskii Non-Degeneracy (IKND), that generalizes a previous non-degeneracy condition introduced by Wall for complex polynomial functions (J. Reine Angew. Math. 509 (1999), 1-19). We prove that IKND is a sufficient condition ensuring that the link of the singularity of $f$ at the origin is smooth and well-defined. We study one-parameter deformations of an IKND map $f$, given by $F(\boldsymbol{x},\varepsilon)=f(\boldsymbol{x})+\theta(\boldsymbol{x},\varepsilon)$, with $ F(0,\varepsilon)=0$. We prove that the deformation is \textit{link-constant} under suitable conditions on $f$ and $\theta$, meaning that the ambient isotopy type of the link remains unchanged along the deformation. Furthermore, by employing a strong version of this non-degeneracy, Strong Inner Khovanskii Non-Degeneracy (SIKND), we obtain results on topological triviality. In the final section, we present link-constant deformations for IKND mixed polynomial functions of two variables. We also explore several applications motivated by the recent findings of Ara\'ujo dos Santos, Bode, and Sanchez Quiceno (Bull. Braz. Math. Soc. (N.S.) 55 (2024), no. 3, Paper No. 34).