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MEASURING THE LOCAL NON-CONVEXITY OF REAL ALGEBRAIC CURVES

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 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.
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Submitted on : Wednesday, August 5, 2020 - 5:00:41 PM
Last modification on : Wednesday, August 5, 2020 - 5:14:13 PM
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Miruna-Stefana Sorea. MEASURING THE LOCAL NON-CONVEXITY OF REAL ALGEBRAIC CURVES. MEGA 2019 - International Conference on Effective Methods in Algebraic Geometry, Jun 2019, Madrid, Spain. ⟨hal-02912362⟩

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