JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact support@jstor.org.. The Johns Hopkins University Press is collaborating with JSTOR to digitize, preserve and extend access to American Journal of Mathematics.Introduction. Let Ro be an ordinary differential domain containing the field of rational numbers, p0: R-AO a differential specialization and L a differential extension of the quotient field K of Ro. Let x E L. It is known that cpo need not extend to a differential specialization from either Ro { x} or Ro { 1/ x} into a differential extension field of AO; indeed, the determination of conditions under which cp0 does so extend has been a subject of persistent interest.This paper, while making some contributions to that investigation (see Propositions 4, 5, 6, and the corollary to Proposition 9), focuses on a related but somewhat different question, one which seems especially pertinent to an attempt at a global treatment of differential algebra using models of differential fields. Relinquishing the condition that T0 extends either to x or to 1/x, we ask whether po even extends to a differential domain whose quotient field is L. The answer provided herein is as follows: A necessary and sufficient condition for po to extend to a differential domain with quotient field L is that Ro is contained in a differential domain R with quotient field L which possesses a proper nonzero differential prime ideal (Theorem 1). An example in Section 5 shows that the condition need not hold; the differential specialization po does extend in the sense mentioned, however, under any one of the following hypotheses: Ro has no minimal differential prime ideal (Theorem 2), the extension LIK is differentially transcendental (Theorem 2), the extension LIK is algebraic (Theorem 3), the extension LIK is generated by an element satisfying a linear differential equation over K (Theorem 4). Thus in particular rp0 extends if LIK is a Picard-Vessiot or Liouvillian extension. Moreover, a necessary and sufficient condition for each member of a chain of differential specializations to extend (Theorem 5) is that the first member of the chain extends.I am grateful to Peter Blum who introduced me to the subject of differential algebra and whose many suggestions have been most helpful.1. Preliminaries. Let R and Ro be differential domains with Ro c R, and zpo:RO-*AO a differential specialization. Then there is a differential field A

No abstract

JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact support@jstor.org.. The Johns Hopkins University Press is collaborating with JSTOR to digitize, preserve and extend access to American Journal of Mathematics.Introduction. The presence of certain anomalies in the theory of differential specializations (as contrasted with the corresponding algebrogeometric theory) has substantially complicated the investigation of ways in which a differential specialization Oo, defined on a differential subring R of a differential field L, may extend or fail to extend differentially to larger subrings of L. More explicitly, if Oo, a differential specialization defined on R, is known to have extensions (not necessarily differential) defined on a subring S of L, it is natural to ask which, if any, of these specializations are in fact differential. The question has proven difficult to answer even if one restricts attention to a single element x of L and seeks merely to determine whether at least one of the extensions of Oo to either x or x-l is differential.The present paper-after reviewing the requisite theory of differential places, and (in Section 2) extending that theory for use in subsequent sections-concentrates on the above question in the case that L is algebraic over the quotient field K of R (assumed to contain the field of rational numbers). Results obtained include the following: If F E R[X] and if FOo ?6 0, then (Theorem 3.1) there is always at least one differential extension of Oo that is defined on every root of F or its reciprocal; for each such root x, moreover, if oo extends to a specialization x -u then (Theorem 3.3) 'O also extends to a differential specialization x -u. In consequence O1 extends differentially to every element of L integral over R, and in fact there is a differential ring RL*, containing the integral closure of R in L, that itself satisfies properties analogous (for differential algebra) to those of an integral closure. Finally, if R is integrally closed and if S is a (not necessarily differential!) subring of L integral over R, Theorem 4.10 guarantees that every specialization extending o/ to S is differential in the sense that it is the restriction to S of a differential specialization defined on RL*.

No abstract