Measuring the local non-convexity of real algebraic curves. (March 2022)

• Record Type: Journal Article
• Authors: Sorea, Miruna-Ştefana
• Is Part Of: Journal of symbolic computation. Volume 109(2022)
• Page Start: 482
• Page End: 509
• Publication Date: 2022-03
• Subjects: Strict local minimum -- Poincaré-Reeb tree -- Non-convexity -- Level curve -- Stabilisation -- Real algebraic curve -- Polar curve -- Star domain -- Smooth Jordan curve; Mathematics -- Data processing -- Periodicals; Numerical analysis -- Data processing -- Periodicals; Automatic programming (Computer science) -- Periodicals; Mathématiques -- Informatique -- Périodiques; Analyse numérique -- Informatique -- Périodiques; Programmation automatique -- Périodiques; Automatic programming (Computer science); Mathematics -- Data processing; Numerical analysis -- Data processing; Periodicals; Electronic journals; 510.285
• DOI: 10.1016/j.jsc.2020.07.017
• Languages: English
• ISSNs: 0747-7171

Abstract

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.