Patchworking the Log-critical locus of planar curves

Abstract

We establish a patchworking theorem à la Viro for the Log-critical locus of algebraic curves in $({\mathbb{C}}^{\ast }{)}^2$. As an application, we prove the existence of projective curves of arbitrary degree with smooth connected Log-critical locus. To prove our patchworking theorem, we study the behaviour of Log-inflection points along families of curves defined by Viro polynomials. In particular, we prove a generalisation of a theorem of Mikhalkin and the second author on the tropical limit of Log-inflection points.

30 pages, 11 figures. Version 2: minor changes, new Remark 3.1, new Figure 6. Comments are welcome