On the Algebraic Closure of Certain Partially Ordered Fields

Abstract

Introduction.This paper is divided into five parts: Part I. Graduated fields.Part II.Fields of meromorphic functions.Part III.Examples for Part II.Part IV.Applications to algebraic differential equations.Part V.Appendix.Part I deals with an abstract algebraic structure called a graduated field, which may be considered as a generalization of a field having a non-Archimedean valuation which is trivial over the prime field.(Cf.Appendix, Note 1.) A graduated field is an ordered quadruple (Jf0, -<, °U, Wq) in which Jf0 is a commutative field of characteristic zero, -< is a partial order relation (cf.Appendix, Note 2) in Jf0, #0 is a subfield ofJf0, ^ is a subgroup of the multiplicative group ¿f (cf. the notational conventions in Appendix, Note 3), and various hypotheses are made about the interrelations of -< with • and 4-in Jf0, in 1^0, and in <% ( § §1,2).In terms of -< an equivalence relation ~ is introduced into Jf by the definition: f~ g means f -g <g.Two subgroups Ji and sé of Jf are defined by setting JÏ equal to the product group "W and by setting s/= {/: 3meJ' such that/~m}.Polynomials whose coefficients lie in a field ¡Fq satisfying Jicz&qc: jrf0 are said to be asymptotically inscribed.For asymptotically inscribed polynomials F the element p of Ji is called a point of stability if q ~ p implies F(q) ~ F(p), a point of instability otherwise.If p is a point of instability for F, its multiplicity is defined to be the minimum s such that p is a point of stability for the sth derivative F(s).The principal result in Part I is Theorem I, §8, which states, in effect, that every graduated field having the following two properties (a) and (b):(a) #0 is algebraically closed, and (b) whenever an asymptotically inscribed polynomial F has a point of instability p, of multiplicity 1, then F has a root y with y ~ p; must also have the following property (c) as well: