An Algebraic Treatment of the Theorem of Closure

Abstract

1. Among the classic theorems concerning the proj ective properties of a pair of conics, perhaps the most interesting is one due to Poncelet, viz. the theorem that if a polygon of n sides can be circumscribed about one conic and at the same time inscribed in a second conic, it is possible to construct an infinite number of such polygons for the given pair of conics. A very elegant demonstration of this theorem may be made by the use of elliptic functions, but a parallel algebraic treatment is also possible. From an algebraic point of view, we have here but one example of a certain interesting class of problems in elimination. We shall mention the general algebraic problem, but shall carry through the details only in the hyperelliptic case. Except in so far as is necessary to make the algebraic steps clear no discussion will be -made of the numerous geometric corollaries that suggest themselves. The present treatment is an attempt to reduce the problem to its simplest form and to prove the theorems needed with a minimum of algebraic machinery. Little emphasis is placed upon the numerous features which serve to individualize the elliptic within the general hyperelliptic problem. The functions considered are those well-known in the transcendental theory, although the methods of proof are of necessity largely new. Constant use has been made of the remarkably clearly written Traite des Fonctions Elliptiques by Halphen. It should be noted that not only are the operations used in this paper algebraic, but that except for a single irrationality, it, every step is essentially rational. Neither the notions of geometric continuity nor of convergence of series are required at any stage. Thus the present discussion is applicable in its entirety to finite fields, a statement which does not hold true of the algebraic treatments already published. Extensive references to the literature on the problem of closure in the elliptic case may be found in the Encyklopddie der Math. Wiss., III, C 1, p. 45 ff., the Encyk. der Geometrie (Simon), p. 105 ff. and in Pascal's Repertorium, IIF, p. 238 ff. Modern algebraic treatments of the Poncelet Polygons are given by 97