Extending differential specializations

Abstract

The study of specializations in differential algebra has been marked by the appearance of several examples which behave very differently from their less complicated brethren in algebraic geometry. In fact, let a denote an ordinary differential field of characteristic zero, U a universal extension of a with field of constants K, with y and z differential indeterminates over U (these notational conventions will be observed from now on). Subscripts will denote derivatives. Then Kolchin has shown that if A=yz'+P(z), where P is a cubic polynomial having distinct roots with coefficients in anK, and if (ti, >) is a generic zero of the general component of A in a {y, z I then O is a differential specialization of ? over a but there is no element aEU such that (0, a) is a differential specialization of (i, t) or (77, D over 5. Another example, which goes back to Ritt [31, shows the existence of another prime differential ideal in a { y, z I whose generic zero (, q) has the property that 0 is a differential specialization of q? over , but not of t1 or of r over a. The situation arising out of the first example was dealt with in [II where it was shown that if R is a local differential domain, meaning that R is local and has a maximal ideal m which is differential, then if t he differential homomorphism 4: R-4R/m cannot be extended either to a or to j, there exist nonnegative integers i, j such that as13,fR. As a corollary it was shown that if a1/aeCR then 4 can be extended either to a differential homomorphism of R[{a or to one of Rtl/aI. In the reference cited a and # were restricted to the differential field of quotients of R, but it is easy to see that this assumption is not necessary. The present paper is concerned with the second of the above-mentioned phenomena and it will be shown that under a certain condition v or t must specialize to 0 over j. I would like to thank Professor Ellis Kolchin and Dr. Jerald Kovacic for their helpful thoughts on this subject. DEFINITION. Let R be a differential subring of U containing the rational numbers, Q. An element f3E U is said to be monic over.R if ,3 is a zero of a differential polynomial of the form y1+f(y) ERR{y where the total degree of f is less than n.