Segre's theorem

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to the definition of a finite oval: tangent, secants, is the order of the projective plane (number of points on a line -1)

In projective geometry Segre's theorem, named after the Italian mathematician Beniamino Segre, is the statement:

This statement was assumed 1949 by the two Finnish mathematicians G. Järnefelt and P. Kustaanheimo and its proof was published in 1955 by B. Segre.

A finite pappian projective plane can be imagined as the projective closure of the real plane (by a line at infinity), where the real numbers are replaced by a finite field K. Odd order means that |K| = n is odd. An oval is a curve similar to a circle (see definition below): any line meets it in at most 2 points and through any point of it there is exactly one tangent. The standard examples are the nondegenerate projective conic sections.

For pappian projective planes of even order there are always ovals which are not conics. In an infinite plane there exist ovals, which are not conics. In the real plane one just glues a half of a circle and a suitable ellipse smoothly.

The proof of Segre's theorem, shown below, uses the 3-point version of Pascal's theorem and a property of a finite field of odd order, namely, that the product of all the nonzero elements equals -1.

Definition of an oval[edit]

  • In a projective plane a set of points is called oval, if:
(1) Any line meets in at most two points.

If the line is an exterior (or passing) line; in case a tangent line and if the line is a secant line.

(2) For any point there exists exactly one tangent at P, i.e., .

For finite planes (i.e. the set of points is finite) we have a more convenient characterization:

  • For a finite projective plane of order n (i.e. any line contains n + 1 points) a set of points is an oval if and only if and no three points are collinear (on a common line).

Pascal's 3-point version[edit]

for the proof is the tangent at
Theorem

Let be an oval in a pappian projective plane of characteristic .
is a nondegenerate conic if and only if statement (P3) holds:

(P3): Let be any triangle on and the tangent at point to , then the points
are collinear.[1]
to the proof of the 3-point Pascal theorem
Proof

Let the projective plane be coordinatized inhomogeneously over a field such that is the tangent at , the x-axis is the tangent at the point and contains the point . Furthermore, we set (s. image)
The oval can be described by a function such that:

The tangent at point will be described using a function such that its equation is

Hence (s. image)

and

I: if is a non degenerate conic we have and and one calculates easily that are collinear.

II: If is an oval with property (P3), the slope of the line is equal to the slope of the line , that means:

and hence
(i): for all .

With one gets

(ii): and from we get
(iii):

(i) and (ii) yield

(iv): and with (iii) at least we get
(v): for all .

A consequence of (ii) and (v) is

.

Hence is a nondegenerate conic.

Remark: Property (P3) is fulfilled for any oval in a pappian projective plane of characteristic 2 with a nucleus (all tangents meet at the nucleus). Hence in this case (P3) is also true for non-conic ovals.[2]

Segre's theorem and its proof[edit]

Theorem

Any oval in a finite pappian projective plane of odd order is a nondegenerate conic section.

3-point version of Pascal's theorem, for the proof we assume
Segre's theorem: to its proof
Proof
[3]

For the proof we show that the oval has property (P3) of the 3-point version of Pascal's theorem.

Let be any triangle on and defined as described in (P3). The pappian plane will be coordinatized inhomogeneously over a finite field , such that and is the common point of the tangents at and . The oval can be described using a bijective function :

For a point , the expression is the slope of the secant Because both the functions and are bijections from to , and a bijection from onto , where is the slope of the tangent at , for we get

(Remark: For we have: )
Hence

Because the slopes of line and tangent both are , it follows that . This is true for any triangle .

So: (P3) of the 3-point Pascal theorem holds and the oval is a non degenerate conic.

References[edit]

  1. ^ E. Hartmann: Planar Circle Geometries, an Introduction to Moebius-, Laguerre- and Minkowski Planes. Skript, TH Darmstadt (PDF; 891 kB), p. 34.
  2. ^ E. Hartmann: Planar Circle Geometries, an Introduction to Moebius-, Laguerre- and Minkowski Planes. Skript, TH Darmstadt (PDF; 891 kB), p. 35.
  3. ^ E. Hartmann: Planar Circle Geometries, an Introduction to Moebius-, Laguerre- and Minkowski Planes. Skript, TH Darmstadt (PDF; 891 kB), p. 41.
  • B. Segre: Ovals in a finite projective plane, Canadian Journal of Mathematics 7 (1955), pp. 414–416.
  • G. Järnefelt & P. Kustaanheimo: An observation on finite Geometries, Den 11 te Skandinaviske Matematikerkongress, Trondheim (1949), pp. 166–182.
  • Albrecht Beutelspacher, Ute Rosenbaum: Projektive Geometrie. 2. Auflage. Vieweg, Wiesbaden 2004, ISBN 3-528-17241-X, p. 162.
  • P. Dembowski: Finite Geometries. Springer-Verlag, 1968, ISBN 3-540-61786-8, p. 149

External links[edit]

  • Simeon Ball and Zsuzsa Weiner: An Introduction to Finite Geometry [1] p. 17.