On a Theorem Related to the Four-Vertex Theorem

Abstract

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On Differentiable Arcs and Curves, VI: Singular Osculating Spaces of Curves of Order n + 1 in Projective n-Space

“…We proved in (6) that the numbers N n pq are bounded for a given n (p + q < n -2) ; and in (5) that (2) £ (n -p)N\ + £ (n-p-l)N n p0 < \n + In. In the present paper we wish to improve (1) and (2) and to show that…”

“…If s moves through B~, then one of the two points t 2 o(s) moves positively through B + while the other one runs negatively through B~, the inflection point being a fixed point. This yields: If the tangential mapping is negative at s, then one of the mappings t t 2…”

“…w _4,o is proper, and the mappings (2) r,_3,o and r-1 n _ 4) o are defined and continuous near s 0 . We label them such that it = it\-z,O(SQ) = it n~1 n-4,o(so) (i = 1, 2).…”

“…We divide the set of these arcs into two classes. B shall belong to the first class if and only if all the mappings (2) it n n -z t o and i t n~1 n -4,o are defined and monotonie on B and have no fixed points in B. Thus any arc B of the second class either contains multiple original points of ir~V-4,o or fixed points of one of the mappings (2).…”

“…B shall belong to the first class if and only if all the mappings (2) it n n -z t o and i t n~1 n -4,o are defined and monotonie on B and have no fixed points in B. Thus any arc B of the second class either contains multiple original points of ir~V-4,o or fixed points of one of the mappings (2). By our induction assumption and the introduction, the number of these points and hence that of the arcs B of the second class is bounded.…”