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arXiv · 1907.08585

Measuring the local non-convexity of real algebraic curves

Miruna-Stefana Sorea
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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.

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Miruna-Stefana Sorea. 2019-07-19. Measuring the local non-convexity of real algebraic curves. https://doi.org/10.1016/j.jsc.2020.07.017

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