Permutations encoding the local shape of level curves of real polynomials via generic projections
Miruna‐Ştefana Sorea · Annales de l’institut Fourier · 2022
The non-convexity of a smooth compact connected component of a real algebraic plane curve can be measured by a combinatorial object: the Poincaré–Reeb tree associated to the curve and to a direction of projection. Here we show that if the chosen projection avoids the bitangents and the inflections to the small enough level curves of a real bivariate polynomial function near a strict local minimum at the origin, then the asymptotic Poincaré–Reeb tree becomes complete binary and its vertices become totally ordered. 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 projection is generic, then the local shape of the curves can be encoded in terms of alternating permutations, called snakes. Snakes offer an effective description of the local geometry and topology, well-suited for computations.