MiS Preprint Repository

Delve into the future of research at MiS with our preprint repository. Our scientists are making groundbreaking discoveries and sharing their latest findings before they are published. Explore repository to stay up-to-date on the newest developments and breakthroughs.

MiS Preprint

Permutations encoding the local shape of level curves of real polynomials via generic projections

Miruna-Ştefana Sorea


The non-convexity of a smooth and compact connected component of a real algebraic plane curve can be measured by a combinatorial object called the Poincare-Reeb tree associated to the curve and to a direction of projection. In this paper we show that if the chosen projection avoids the bitangents and the inflectional tangencies to the small enough level curves of a real bivariate polynomial function near a strict local minimum at the origin, then the asymptotic Poincare-Reeb tree becomes a complete binary tree and its vertices become endowed with a total order relation. Such a projection direction is called generic. We prove that for any such asymptotic family of level curves, there are finitely many intervals on the real projective line outside of which all the directions are generic with respect to all the curves in the family. If the choice of the direction of projection is generic, then the local shape of the curves can be encoded in terms of alternating permutations, that we call snakes. The snakes offer an effective description of the local geometry and topology, well-suited for further computations.

Oct 29, 2019
Nov 3, 2019
MSC Codes:
14P25, 14P05, 14H20, 14B05, 05C0, 14Q05, 26C, 58K
generic projection, real algebraic curve, strict local minimum, polar curve, permutation

Related publications

2022 Journal Open Access
Miruna-Stefana Sorea

Permutations encoding the local shape of level curves of real polynomials via generic projections

In: Annales de l'Institut Fourier, 72 (2022) 4, pp. 1661-1703