Regions surrounded by circles whose Poincar\'e-Reeb graphs are trees

TL;DR

This paper investigates regions in the plane bounded by circles with Poincaré-Reeb graphs that are trees, exploring their structure and properties through inductive rules, contributing to understanding algebraic maps onto such regions.

Contribution

The study characterizes regions bounded by circles with tree-structured Poincaré-Reeb graphs and introduces an inductive method to analyze their topological features.

Findings

Poincaré-Reeb graphs of these regions are trees under certain conditions.

An inductive rule generates the trees from a disk in the plane.

The work advances understanding of algebraic maps onto prescribed regions.

Abstract

Regions in the Euclidean plane surrounded by circles are fundamental geometric and combinatorial objects. Related studies have been done and we cannot explain them precisely, or roughly, well. We study such regions whose Poincar\'e-Reeb graphs are trees and investigate the trees obtained by a certain inductive rule from a disk in the plane. The Poincar\'e-Reeb graph of such a region is a graph whose underlying set is the set of all components of level sets of the restriction of the canonical projection to the closure and whose vertices are points corresponding to the components containing {\it singular} points. Related studies were started by the author, motivated by importance and difficulty of explicit construction of a real algebraic map onto a prescribed closed region in the plane.