Measuring the local non-convexity of real algebraic curves
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Submission date: 22. Jul. 2019
MSC-Numbers: 14P25, 14P05, 14H20, 14B05, 05C05, 14Q05, 26C, 58K
Keywords and phrases: strict local minimum, Poincare-Reeb tree, non-convexity, level curve, stabilisation, real algebraic curve, polar curve, star domain
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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.