Complete models of differential fields

Abstract

Introduction. This work constitutes a step in the direction of a global theory of differential algebra. The main result (Theorem 1 of ?2) is the following: Let K be an ordinary differential field of characteristic zero. Then there is a premodel of K with the property that every maximal differential homomorphism from a differential subring of K, whose differential field of quotients is K, induces a point on this pre-model. Under a certain condition these points will be uniquely determined. It is not yet known whether there exist differential fields of the form k , where the xl belong to some universal differential field extension of k, which do not admit complete models over k. Even in the case where x1, ..., x5 are differentially algebraically independent the answer is not apparent. For definitions, see the discussion preceding the above-mentioned Theorem 1. The author would like to thank Professors A. Seidenberg and E. R. Kolchin for their helpful comments and suggestions. NOTATION AND CONVENTIONS. We denote the integers, rationals, reals, and complex numbers by Z, Q, R, C, respectively, and usually consider Z as a differential ring, and Q, R, and C as differential fields of constants. If R is an integral domain, we write qf (R) for its quotient field. If S is a differential integral domain, S{y} will denote the ring of differential polynomials with coefficients in S. The letter y will represent a differential indeterminate, while x,, z will stand for differential quantities which may or may not be differential indeterminates. We begin by examining some of the elementary properties of the category D of differential rings and differential homomorphisms (these are ordinary differential rings, although many of the arguments remain valid in the partial case). For the moment, no restriction is placed on zero divisors or characteristic. First of all, D admits coproducts in the sense that if A, B E D we can define A 0 B by taking the usual tensor product of A and B (considered as Z-algebras) and setting (a 0 b)'=a' X b+a X b'. This makes A X B into a differential ring satisfying the conditions for a coproduct in a category. In fact, we have differential