Manifolds of difference polynomials

1. It is the purpose of this paper to develop in some detail the structure of the manifolds determined by systems of difference polynomials. Our results will necessarily be confined to the case of polynomials in an abstract field, since a suitable existence theorem for analytic difference equations is not available. The ideal theory, developed by J. F. Ritt and H. W. Raudenbush(1) for abstract systems of difference polynomials, is therefore fundamental in our work. 2. In Part I of our paper we describe a theoretical method for elimination of unknowns in systems of algebraic difference equations. We employ this method to prove analogues for difference fields of fundamental theorems of algebra on field extensions. With the aid of these results we show in Theorem III that the number of arbitrary(2) unknowns in a prime difference ideal is constant for all possible choices of sets of arbitrary unknowns. 3. Part II is concerned with the manifold of a single algebraically irreducible difference polynomial in an abstract field. A factorization process for polynomials in analytic fields was developed by J. F. Ritt(3) in determinining the maximum number of irreducible manifolds, not held by polynomials of zero order, in the decomposition of the manifold of a first order difference polynomial. In Theorem IV we show that, when the Ritt factorization process is applied to a polynomial A in an abstract field, each of the polynomial sequences it produces actually determines a prime ideal held by A but not by any polynomial of lower order than A. Furthermore, all such prime ideals are obtained in this way. This constitutes a form of existence theorem for difference polynomials in abstract fields, and is fundamental in the further development of the theory. The irreducible manifolds of A determined by the factorization process we call the general solution of A. We shall see that, if A is of first order, all solutions are included in the general solution. This result confirms, in a gen-