{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,1]],"date-time":"2026-03-01T13:19:48Z","timestamp":1772371188664,"version":"3.50.1"},"reference-count":11,"publisher":"Cambridge University Press (CUP)","issue":"1","license":[{"start":{"date-parts":[[2014,3,12]],"date-time":"2014-03-12T00:00:00Z","timestamp":1394582400000},"content-version":"unspecified","delay-in-days":13160,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1978,3]]},"abstract":"In this paper, we show that the theory of ordered differential fields has a model completion. We also show that any real differential field, finitely generated over the rational numbers, is isomorphic to some field of real meromorphic functions. In the last section of this paper, we combine these two results and discuss the problem of deciding if a system of differential equations has real analytic solutions. The author wishes to thank G. Stengle for some stimulating and helpful conversations and for drawing our attention to fields of real meromorphic functions.<\/jats:p>\u00a7 1. Real and ordered fields.<\/jats:bold> A real field<\/jats:italic> is a field in which \u22121 is not a sum of squares. An ordered field<\/jats:italic> is a field F<\/jats:italic> together with a binary relation < which totally orders F<\/jats:italic> and satisfies the two properties: (1) If 0 < x<\/jats:italic> and 0 < y<\/jats:italic> then 0 < xy<\/jats:italic>. (2) If x<\/jats:italic> < y<\/jats:italic> then, for all z<\/jats:italic> in F<\/jats:italic>, x<\/jats:italic> + z<\/jats:italic> < y<\/jats:italic> + z<\/jats:italic>. An element x<\/jats:italic> of an ordered field is positive if x<\/jats:italic> > 0. One can see that the square of any element is positive and that the sum of positive elements is positive. Since \u22121 is not positive, an ordered field is a real field. Conversely, given a real field F<\/jats:italic>, it is known that one can define an ordering (not necessarily uniquely) on F<\/jats:italic> [2, p. 274]. An ordered field F<\/jats:italic> is a real closed field<\/jats:italic> if: (1) every positive element is a square, and (2) every polynomial of odd degree with coefficients in F<\/jats:italic> has a root in F<\/jats:italic>. For example, the real numbers form a real closed field. Every ordered field can be embedded in a real closed field. It is also known that, in a real closed field K<\/jats:italic>, polynomials satisfy the intermediate value property, i.e. if f(x)<\/jats:italic> \u2208 K[x]<\/jats:italic> and a, b<\/jats:italic> \u2208 K, a<\/jats:italic> < b<\/jats:italic>, and f(a)f(b)<\/jats:italic> < 0 then there is a c<\/jats:italic> in K<\/jats:italic> such that f(c)<\/jats:italic> = 0.<\/jats:p>","DOI":"10.2307\/2271951","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T21:46:34Z","timestamp":1146951994000},"page":"82-91","source":"Crossref","is-referenced-by-count":34,"title":["The model theory of ordered differential fields"],"prefix":"10.1017","volume":"43","author":[{"given":"Michael F.","family":"Singer","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200049914_ref010","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9939-1958-0093655-0"},{"key":"S0022481200049914_ref011","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9939-1969-0248122-5"},{"key":"S0022481200049914_ref009","first-page":"174","article-title":"Some basic theorems in differential algebra (characteristic p, arbitrary)","volume":"73","author":"Seidenberg","year":"1952","journal-title":"Transactions of the American Mathematical Society"},{"key":"S0022481200049914_ref001","unstructured":"Blum L. , Generalized algebraic structures: A model theoretic approach, Ph. 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Thesis, Massachusetts Institute of Technology, 1968."},{"key":"S0022481200049914_ref002","first-page":"322","volume-title":"Rendiconti della Academia Nazionale Dei Lincei","volume":"59","author":"Lacava","year":"1975"},{"key":"S0022481200049914_ref003","volume-title":"Algebra","author":"Lang","year":"1970"},{"key":"S0022481200049914_ref004","first-page":"354","article-title":"Enumerable sets are Diophantine","volume":"11","author":"Matijasevic","year":"1970","journal-title":"Soviet Mathematics, Doklady"},{"key":"S0022481200049914_ref005","volume-title":"Analysis on real and complex manifolds","author":"Narasimhan","year":"1968"},{"key":"S0022481200049914_ref006","first-page":"113","volume-title":"Bulletin of the Research Council of Israel","author":"Robinson","year":"1959"},{"key":"S0022481200049914_ref008","volume-title":"Saturated model theory","author":"Sacks","year":"1972"},{"key":"S0022481200049914_ref007","doi-asserted-by":"publisher","DOI":"10.1016\/0097-3165(73)90009-5"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200049914","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,27]],"date-time":"2019-05-27T20:19:54Z","timestamp":1558988394000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200049914\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1978,3]]},"references-count":11,"journal-issue":{"issue":"1","published-print":{"date-parts":[[1978,3]]}},"alternative-id":["S0022481200049914"],"URL":"https:\/\/doi.org\/10.2307\/2271951","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1978,3]]}}}