Poincar\'e-Reeb graphs of real algebraic domains

TL;DR

This paper introduces Poincaré-Reeb graphs for real algebraic domains, showing how they encode the shape of these domains and characterizing which graphs can be realized as such.

Contribution

It provides a characterization of Poincaré-Reeb graphs for algebraic domains and demonstrates realizability conditions for certain transversal graphs.

Findings

Poincaré-Reeb graphs encode algebraic domain shapes.

Any transversal graph with vertices of valency 1 or 3 on distinct lines can be realized.

The study links algebraic geometry with topological graph representations.

Abstract

An algebraic domain is a closed topological subsurface of a real affine plane whose boundary consists of disjoint smooth connected components of real algebraic plane curves. We study the geometric shape of an algebraic domain by collapsing all vertical segments contained in it: this yields a Poincar\'e-Reeb graph, which is naturally transversal to the foliation by vertical lines. We show that any transversal graph whose vertices have only valencies 1 and 3 and are situated on distinct vertical lines can be realized as a Poincar\'e-Reeb graph.