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 ( ℂ ∗ stretchy="false">) 2 {(\mathbb{C}^{\ast})^{2}} . As an application, we prove existence projective arbitrary degree with smooth connected locus. To our theorem, study behaviour Log-inflection points along families defined by polynomials. In particular, generalisation Mikhalkin and second author on tropical limit points.