On the structure of differential polynomials and on their theory of ideals

In the first part of this paper a special class of differential ideals(') is investigated. The results of this section are used in the following one to derive some structural properties of differential polynomials. The last part of the paper is devoted to a special differential ideal. With the help of some conventions of notation, more precise indications of the scope of our work may be given. Let IR. denote the ring of differential polynomials, with rational numbers for coefficients, in the unknown y. The special class of differential ideals studied in Part I is composed of those generated by yP, where p is a positive integer. These ideals are among the most simple ideals encountered in the theory of differential equations. Viewed as algebraic entities, however, they are by no means trivial. We denote the ith derivative of y by yi; R. thus appears as a polynomial ring with infinitely many indeterminates y, Yl, Y2, * . Since the Hilbert basis theorem does not hold on 'A, one would expect almost any ideal in R. to be unruly. By introducing order relations into fR we have been able to proceed despite the absence of the basis theorem and to obtain fairly comprehensive results concerning these differential ideals. In particular a simple criterion for determining the membership in such an ideal of an element of R, is obtained which plays a fundamental role in Part II. This second part establishes the abstract counterparts of some results of J. F. Ritt concerning essential manifolds which figure in the decomposition of a manifold into irreducible ones. It has been found possible to present results which cover situations not discussed by him. The differential ideal discussed in Part III is that generated by uv, where u and v are unknowns. Among other properties, it is shown that this ideal has no representation as the intersection or product of two differential ideals, whose manifolds are respectively u = 0 with v arbitrary, and v = 0 with u arbitrary. This result owes its interest to the fact that the manifold of the equation uv=O is evidently reducible into the union of the two manifolds just defined. In a narrow sense, this paper is independent of other literature; the argu-