Tree-like Curves and Their Number of Inflection Points

In this short note we give a criterion when a planar tree-like curve, i.e. a generic curve in R 2 each double point of which cuts it into two disjoint parts, can be send by a diffeomorphism of R 2 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 R 2. §1. Introduction This paper provides a partial answer to the following question posed to the author by V.Arnold in June 95. Given a generic immersion c : S 1 → R 2 (i.e. with double points only) let ♯ inf (c) denote the number of inflection points on c (assumed finite) and let [c] denote the class of c, i.e. the connected component in the space of generic immersions of S 1 to R 2 containing c. Finally, let ♯ inf [c] = min c ′ ∈[c] ♯ inf (c ′). Problem. Estimate ♯ inf [c] in terms of the combinatorics of c. The problem itself is apparently motivated by the following classical result due to Möbius, see e.g. [Ar3]. Theorem. Any embedded noncontractible curve on RP 2 has at least 3 inflection points. The present paper contains some answers for the case when c is a tree-like curve, i.e. satisfies the condition that if p is any double point of c then c \ p has 2 connected components. We plan to drop the restrictive assumption of tree-likeness in our next paper, see [Sh]. Classes of tree-like curves are naturally enumerated by partially directed trees with a simple additional restriction on directed edges, see §2. It was a pleasant surprise that for the classes of tree-like curves there 1991 Mathematics Subject Classification. Primary 53A04. 2 B. SHAPIRO exists a (relatively) simple combinatorial criterion characterizing when [c] contains a nonflattening curve, i.e. ♯ inf [c] = 0 in terms of its tree. (Following V.Arnold we use the word 'nonflattening' in this text as the synonym for the absence of inflection points.) On the other hand, all attempts to find a closed formula for ♯ inf [c] in terms of partially directed trees failed. Apparently such a formula does not exist, see the Concluding Remarks. The paper is organized as follows. §2 contains some general information on tree-like curves. §3 contains a criterion of noflattening. §4 presents some upper and …