An existence theorem in the algebraic study of homogeneous linear ordinary differential equations

Abstract

Introduction. The Picard-Vessiot theory is a Galois theory of homogeneous linear ordinary differential equations. Unlike ordinary Galois theory, however, in which fixing a universe automatically determines the zeros of a given polynomial, Picard-Vessiot theory is complicated by the fact that a given homogeneous linear ordinary differential polynomial L(y) of order ?1 possesses infinitely many zeros. To simplify matters, therefore, one wishes to choose those zeros of L(y) which are the simplest in a certain sense. More specifically, it is seen in a paper (to appear) generalizing the PicardVessiot theory as presented by Kolchin [1] that the algebraic and topological structures involved in that generalization are strongly dependent on the constants introduced when one adjoins zeros of L(y) to the ground field. It is desirable, therefore, to choose zeros of L(y) in such a manner that the constants introduced by these zeros are subject to certain restrictive conditions. This requires a theorem asserting the existence of zeros of L(y) for which these restrictive conditions are satisfied. The main result of this paper is a theorem on the existence of this special type of zeros of L(y). We fix an abstract ordinary differential field F of characteristic zero and with field of constants C (i.e. F is an abstract field of characteristic zero in which a derivation is defined, and C is the set of all elements of F which have derivative equal to 0). We shall assume throughout that we are given Q, a universal differential extension field of F, the existence of which is proved in [3, Chap. I, ?5]. This universal extension will contain all the elements which enter the discussion. In particular, Q will contain an algebraic closure C of C. We use the standard notation for derivatives; thus, for a EQ the successive derivatives of a are denoted a', a, , ( . . For the adjunction of elements to differential fields (for example, to the differential field F), we shall use F( . . . ) to denote ordinary field adj unction, F { ... } to denote differential ring adjunction, and F( ... ) to denote differential field adjunction: thus, for example,