A special prepared system for two quadratics in N variables

Abstract

A complete system of concomitants for two quadratics in n variables is a system of concomitants, in terms of which every rational integral concomitant of the two quadratics may be expressed rationally and integrally. The existence of such a finite complete system is a particular case of Gordan's theorem. There are three possible types of complete systems:

(1) G = G(a,r,u₁,u₂, . . . u₂ uₙ₋₂,x); (2) H = G(a,r,x₁,x₂, . . . xₙ₋₁,xₙ); (3) K = K(a,r,π₁,π₂, . . . π₋₁,x)

where each contains co-efficients a, r of the two quadratic ground forms, u₁ denote plane co-ordinates, x and x₁ denote point co-ordinates, and π₁ denote general line co-ordinates (See part I. § I). From a geometrical point of view the K system is the most important but also it is the largest and most difficult to determine. This system K is known for the cases n=-2,² n= 3,³ and n = 4.⁴ The number of irreducible concomitants for the cases n = 2, n = 3, and n = 4 are respectively 6, 20 and 122. When n=2 or 3, the system is strictly irreducible and thus the complete system in these two cases is also the minimum system.The G system has been determined and in a joint paper by H. W. Turnbull and the author, to be published shortly in the Proceedings of the Royal Society of Edinburgh, a complete system is found, when the co-ordinates π (or p₁) are decomposed into their components u₁ and x₁.In this paper we are interested in the K system and, though no complete system is determined in the general case, a distinct step is made in that direction. If every concomitant of the two quadratics may be expressed as a product of symbolic factors, where each symbol occurs an even number of times in each product and distinguishing marks on equivalent symbols may be neglected, the totality of such factors is said to form a prepared system for the two quadratics. In part I we determine a prepared system, in terms of which every concomitant of the two quadratics, if multiplied by a suitable invariant factor, may be expressed. This prepared

system is comparatively simple and consists of 2ⁿ-1 factors. In obtaining a prepared system, in terms of which every concomitant can be expressed, without being multiplied by an invariant factor, new more complicated bracket factors must be introduced, when n ≥4. The number of such new bracket factors is 1, 8, and 52 for the cases n=4, 5, and 6 respectively.