2015/04/27 by Lionel Lang, Lang, Lionel
Computer Science · Mathematics · #14P25 #14T05 #Advanced Differential Equations and Dynamical Systems #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Polynomial and algebraic computation
paper · pdf · doi:10.48550/arxiv.1504.07256
openalex publication_date 2015/04/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this paper, we suggest the following generalisation of Mikhalkin's simple Harnack curves: a generalised simple Harnack curve is a parametrised real algebraic curve in (ℂ^*)2 with totally real logarithmic Gauss map. We investigate which of the many properties of simple Harnack curves survive the latter generalisation. We also show how tropical geometry allows to construct plenty of examples. Since generalised Harnack curves can develop arbitrary singularities, in contrast with the original definition where only real isolated double points can appear, we pay a special attention to the simplest new instance of generalised Harnack curves, namely curves with a single hyperbolic node. In particular, we give their topological classification as in \citeMikh and show how such curves can be recovered from their spine.