Segre's theorem

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.

In pappian projective planes of even order greater than four there are 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.

is an exterior (or passing) line; in case

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

an oval in a pappian projective plane of characteristic

is a nondegenerate conic if and only if statement (P3) holds: Let the projective plane be coordinatized inhomogeneously over a field

, the x-axis is the tangent at the point

(s. image) The oval

such that its equation is Hence (s. image) I: if

is a non degenerate conic we have

and one calculates easily that

is an oval with property (P3), the slope of the line

is equal to the slope of the line

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.

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

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

The pappian plane will be coordinatized inhomogeneously over a finite field

is the common point of the tangents at

can be described using a bijective function

) Hence Because the slopes of line

This is true for any triangle

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

to the definition of a finite oval: $t$ tangent, $s_{1},...s_{n}$ secants, $n$ is the order of the projective plane (number of points on a line -1)

{\displaystyle t} — to the definition of a finite oval: $t$ tangent, $s_{1},...s_{n}$ secants, $n$ is the order of the projective plane (number of points on a line -1)