Skeleta of affine hypersurfaces

Abstract

A smooth affine hypersurface $Z$ of complex dimension $n$ is homotopy equivalent to an $n$–dimensional cell complex. Given a defining polynomial $f$ for $Z$ as well as a regular triangulation ${\mathcal{T}}_{△}$ of its Newton polytope $△$, we provide a purely combinatorial construction of a compact topological space $S$ as a union of components of real dimension $n$, and prove that $S$ embeds into $Z$ as a deformation retract. In particular, $Z$ is homotopy equivalent to $S$.