Poincaré-Reeb graphs of real algebraic domains

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 Poincaré-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 Poincaré-Reeb graph.

31 pages, 37 figures. We improved the exposition, according to the referee's suggestions. We made small changes, added references and a question from the reviewers. To appear in Revista Matemática Complutense

References in corpus (3)

• Measuring the local non-convexity of real algebraic curves
• Constructing Separable Arnold Snakes of Morse Polynomials
• Permutations encoding the local shape of level curves of real polynomials via generic projections