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 fieldK. 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 ellipsesmoothly.
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.
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)
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:
(i): for all .
With one gets
(ii): and from we get
(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.
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.
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:
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.