Segre's theorem

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)

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

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

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.