Geometrische Ordnungen

Independence structures in the geometry of orders

“…In some of these cases there are analogues of the Four-vertex Theorem; i.e., results on the minimum number of Of'-singular points of a JU-differentiable curve of Jd-order > k. Also the classification of the Jd-differentiable curves of~-order (k + 1) is known in some cases [2], [16]. In both of these topics more research seems desirable.…”

“…In the local theory, a point p on B, the base point, is introduced, and the behaviour of B near p is examined. For details we refer to the first pages of [2]. However, the following concepts are central: The #g-order of B is the number sup ] K n B I.…”

“…An Arc [A curve] is the homeomorphic image of a segment [of the projective line].The geometry of orders of arcs and curves is developed in [2]. In a fundamental domain G, a set #g of curves and arcs K is given.…”

Differentiability conditions and bounds on singular points

“…That a strongly differentiable ([3], 3.1) J^ contains at most four singular points can be found in 4.1.4.3 of [1] and in 3.6 of [4].…”

“…This characterization is similar to that of 0. Haupt and H. Kϋnneth for linearly singular points of arcs of linear order three ( [1], 3.2.1).…”