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 Poincaré-Reeb graph, whose role is to encode the shape of such curves and allow us to quantify their non-convexity. Moreover, we prove that in this setting the Poincaré-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.

本文的目标是测量实代数平面曲线的紧凑光滑连通分量的非凸性。我们首先在一般环境中研究这些曲线，然后在渐近环境中研究这些曲线。特别地，我们考虑在实仿射平面原点的严格局部最小值的足够小邻域中的实双变量多项式的足够小级别。我们介绍并描述了一种新的组合对象，称为 Poincaré-Reeb 图，其作用是对此类曲线的形状进行编码，并允许我们量化它们的非凸性。此外，我们证明在这种情况下，庞加莱-里布图是一棵平面树，可以用作研究严格局部最小值附近的水平曲线的渐近行为的工具。最后，使用实极曲线，