On a type of algebraic differential manifold

Abstract

The manifolds(') to be investigated, which are manifolds of systems of differential polynomials in a single unknown, possess a degree of analogy to bounded sets of numbers. They are manifolds which may be said to contain infinity as a solution; more definitely, zero is not a limit of reciprocals of solutions. For manifolds of this type, which will be called limited, operations of addition, multiplication and differentiation will be studied. Given two manifolds(2) 91k and 92, their arithmetic sum is secured by completing into a manifold the totality of functions each of which is, in some area, the sum of a solution in 9Nh And a solution in 9)2. Multiplication is defined similarly. It turns out that if 9Th and 9)2 are general solutions of equations of the first order, and are limited, their sum and product are limited. On the other hand, as is shown by examples based on the theory of the elliptic functions, when 9Th and 9N2 involve more than one arbitrary constant their limited character may not be communicated to their sum and product; what is equivalent to this, as far as multiplication is concerned, is the rather unexpected result that the product of two manifolds may contain zero even if neither manifold does. The derivative of a limited manifold proves to be limited in all cases.