Poincaré-Reeb Graphs of Real Algebraic Domains

Abstract: Consider a real bivariate polynomial function that has a strict local minimum at the origin and that vanishes at this point. In a sufficiently small neighborhood of the origin, the non-zero level curves of this function are smooth Jordan curves. Whenever the origin is a Morse strict local minimum, the small enough level curves become boundaries of convex topological disks. Otherwise, the levels may be non-convex, as was proven by M. Coste. In order to measure this non-convexity, we introduce a combinatorial object called the Poincaré-Reeb tree associated to a level curve and to a projection direction. Our goal is to characterize all topological types of Poincaré-Reeb trees. I will explain how to construct a family of polynomials that realizes a large class of these trees. Moreover, in a joint work with Arnaud Bodin and Patrick Popescu-Pampu, we extend the previous method of study of non-convexity to real algebraic domains.

computational geometrydiscrete mathematicscommutative algebracombinatorics

Audience: researchers in the topic

Copenhagen-Jerusalem Combinatorics Seminar

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