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

TL;DR

This paper introduces a method to encode the local shape of level curves of real polynomials using permutations called snakes, based on generic projections that simplify the Poincare-Reeb tree structure.

Contribution

It establishes a link between generic projections of polynomial level curves and combinatorial objects called snakes, providing a new way to analyze local curve geometry.

Findings

Asymptotic Poincare-Reeb trees become complete binary trees under generic projections.

The local shape of curves can be encoded using alternating permutations (snakes).

Finitely many non-generic directions exist for a family of level curves.

Abstract

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.