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

Julian D. Espinel Leal; Eder L. Sanchez Quiceno

Paper

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

Abstract

For real polynomial maps and mixed polynomial maps

, with

, we introduce the notion of \textit{Inner Khovanskii Non-Degeneracy} (IKND), which generalize a previous non-degeneracy condition for complex polynomial functions introduced by Wall Wall (J. Reine Angew. Math. 509 (1999), 1-19.). We prove that IKND is a sufficient condition that ensures the link of the singularity of

at the origin is smooth and well-defined. We then study one-parameter deformations of an IKND map

, given by

. We prove that the deformation is \textit{link-constant} under suitable conditions on

and

, meaning that the ambient isotopy type of the link remains unchanged along the deformation. Furthermore, by employing a strong version of this non-degeneracy, \textit{Strong Inner Khovanskii Non-Degeneracy} (SIKND), we obtain results on topological triviality. Finally, as an application of our link-constancy results, we provide a complete characterization of the links arising from IKND mixed polynomial functions in two variables, thereby extending and refining the characterization previously established by Bode (Bull. Braz. Math. Soc. (N.S.) 56 (2025), no. 4, Paper No. 55).