Tree-like curves and their inflection points

TL;DR

This paper establishes criteria for transforming planar tree-like curves into inflection-free curves via diffeomorphisms and provides bounds on the minimal number of inflection points based on combinatorial properties.

Contribution

It introduces a criterion for when a tree-like curve can be diffeomorphically transformed to have no inflection points and offers bounds related to their combinatorial structure.

Findings

Criteria for inflection point elimination in tree-like curves

Upper and lower bounds for minimal inflection points

Relation between combinatorics and inflection points

Abstract

We give a criterion when a planar tree-like curve, i.e. a generic immersed plane curve each double point of which cuts it into two disjoint parts, can be send by a diffeomorphism of the plane onto a curve with no inflection points. We also present some upper and lower bounds for the minimal number of inflection points on such curves unremovable by diffeomorphisms of the plane in terms of their combinatorics.