# Measuring the local non-convexity of real algebraic curves

## TL;DR

This paper introduces a new combinatorial tool called the Poincare-Reeb graph to quantify the non-convexity of real algebraic plane curves, analyzing their local and asymptotic shapes near minima.

## Contribution

It develops the Poincare-Reeb graph as a novel method to encode and study the shape and non-convexity of algebraic curves, especially near local minima.

## Key findings

- Poincare-Reeb graph is a plane tree for these curves.
- The shape of level curves stabilizes near local minima.
- No spiralling phenomena occur near the origin.

## Abstract

The goal of this paper is to measure the non-convexity of compact and smooth connected components of real algebraic plane curves. We study these curves first in a general setting and then in an asymptotic one. In particular, we consider sufficiently small levels of a real bivariate polynomial in a small enough neighbourhood of a strict local minimum at the origin of the real affine plane. We introduce and describe a new combinatorial object, called the Poincare-Reeb graph, whose role is to encode the shape of such curves and to allow us to quantify their non-convexity. Moreover, we prove that in this setting the Poincare-Reeb graph is a plane tree and can be used as a tool to study the asymptotic behaviour of level curves near a strict local minimum. Finally, using the real polar curve, we show that locally the shape of the levels stabilises and that no spiralling phenomena occur near the origin.