Good real pictures of corank one map germs from the $n$-space to the $(n+1)$-space

Abstract

We study corank one $A$-finite germs $f:(\mathbb{R}^n,0)\rightarrow (\mathbb{R}^{n+1},0)$ and their complexifications. More precisely, we study when these germs provide good real pictures of the complex germs, i.e., when there is a real deformation that has the same homology in the image (hence, homotopy) than the generic complex deformation. We give a new sufficient condition that can be computed in practice, as well as examples.