On model complete differential fields
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- by E. Hrushovski and M. Itai PDF
- Trans. Amer. Math. Soc. 355 (2003), 4267-4296 Request permission
Abstract:
We develop a geometric approach to definable sets in differentially closed fields, with emphasis on the question of orthogonality to a given strongly minimal set. Equivalently, within a family of ordinary differential equations, we consider those equations that can be transformed, by differential-algebraic transformations, so as to yield solutions of a given fixed first-order ODE $X$. We show that this sub-family is usually definable (in particular if $X$ lives on a curve of positive genus). As a corollary, we show the existence of many model-complete, superstable theories of differential fields.References
- James Ax, On Schanuel’s conjectures, Ann. of Math. (2) 93 (1971), 252–268. MR 277482, DOI 10.2307/1970774
- Alexandru Buium, Effective bound for the geometric Lang conjecture, Duke Math. J. 71 (1993), no. 2, 475–499. MR 1233446, DOI 10.1215/S0012-7094-93-07120-7
- Alexandru Buium, Differential algebraic groups of finite dimension, Lecture Notes in Mathematics, vol. 1506, Springer-Verlag, Berlin, 1992. MR 1176753, DOI 10.1007/BFb0087235
- G. Cherlin and S. Shelah, Superstable fields and groups, Ann. Math. Logic 18 (1980), no. 3, 227–270. MR 585519, DOI 10.1016/0003-4843(80)90006-6
- Robin Hartshorne, Algebraic geometry, Graduate Texts in Mathematics, No. 52, Springer-Verlag, New York-Heidelberg, 1977. MR 0463157, DOI 10.1007/978-1-4757-3849-0
- Ehud Hrushovski, Proof of Manin’s theorem by reduction to positive characteristic, Model theory and algebraic geometry, Lecture Notes in Math., vol. 1696, Springer, Berlin, 1998, pp. 197–205. MR 1678551, DOI 10.1007/978-3-540-68521-0_{1}1
- E. Hrushovski, ODE’s of order 1 and a generalization of a theorem of Jouanolou (to appear).
- Ehud Hrushovski, Almost orthogonal regular types, Ann. Pure Appl. Logic 45 (1989), no. 2, 139–155. Stability in model theory, II (Trento, 1987). MR 1044121, DOI 10.1016/0168-0072(89)90058-4
- E. Hrushovski, Ž. Sokolovic, Strongly minimal sets in differentially closed fields, to appear in Transactions of the AMS
- Albert Eagle, Series for all the roots of the equation $(z-a)^m=k(z-b)^n$, Amer. Math. Monthly 46 (1939), 425–428. MR 6, DOI 10.2307/2303037
- E. R. Kolchin, Constrained extensions of differential fields, Advances in Math. 12 (1974), 141–170. MR 340227, DOI 10.1016/S0001-8708(74)80001-0
- E. R. Kolchin, Algebraic groups and algebraic dependence, Amer. J. Math. 90 (1968), 1151–1164. MR 240106, DOI 10.2307/2373294
- Serge Lang, Abelian varieties, Springer-Verlag, New York-Berlin, 1983. Reprint of the 1959 original. MR 713430, DOI 10.1007/978-1-4419-8534-7
- Serge Lang, Algebra, Addison-Wesley Publishing Co., Inc., Reading, Mass., 1965. MR 0197234
- Serge Lang, Introduction to algebraic geometry, Addison-Wesley Publishing Co., Inc., Reading, Mass., 1972. Third printing, with corrections. MR 0344244
- Angus Macintyre, On $\omega _{1}$-categorical theories of fields, Fund. Math. 71 (1971), no. 1, 1–25. (errata insert). MR 290954, DOI 10.4064/fm-71-1-1-25
- David Marker, Model theory of differential fields, Model theory, algebra, and geometry, Math. Sci. Res. Inst. Publ., vol. 39, Cambridge Univ. Press, Cambridge, 2000, pp. 53–63. MR 1773702
- Anand Pillay, Geometric stability theory, Oxford Logic Guides, vol. 32, The Clarendon Press, Oxford University Press, New York, 1996. Oxford Science Publications. MR 1429864
- Anand Pillay, Differential Galois theory. II, Ann. Pure Appl. Logic 88 (1997), no. 2-3, 181–191. Joint AILA-KGS Model Theory Meeting (Florence, 1995). MR 1600903, DOI 10.1016/S0168-0072(97)00021-3
- Bruno Poizat, Groupes stables, Nur al-Mantiq wal-Maʾrifah [Light of Logic and Knowledge], vol. 2, Bruno Poizat, Lyon, 1987 (French). Une tentative de conciliation entre la géométrie algébrique et la logique mathématique. [An attempt at reconciling algebraic geometry and mathematical logic]. MR 902156
- Bruno Poizat, Une théorie de Galois imaginaire, J. Symbolic Logic 48 (1983), no. 4, 1151–1170 (1984) (French). MR 727805, DOI 10.2307/2273680
- Maxwell Rosenlicht, Extensions of vector groups by abelian varieties, Amer. J. Math. 80 (1958), 685–714. MR 99340, DOI 10.2307/2372779
- Maxwell Rosenlicht, The nonminimality of the differential closure, Pacific J. Math. 52 (1974), 529–537. MR 352068, DOI 10.2140/pjm.1974.52.529
- Gerald E. Sacks, Saturated model theory, Mathematics Lecture Note Series, W. A. Benjamin, Inc., Reading, Mass., 1972. MR 0398817
- Jean-Pierre Serre, Groupes algébriques et corps de classes, Publications de l’Institut de Mathématique de l’Université de Nancago, No. VII, Hermann, Paris, 1975 (French). Deuxième édition. MR 0466151
- Saharon Shelah, Uniqueness and characterization of prime models over sets for totally transcendental first-order theories, J. Symbolic Logic 37 (1972), 107–113. MR 316239, DOI 10.2307/2272553
- Saharon Shelah, Differentially closed fields, Israel J. Math. 16 (1973), 314–328. MR 344116, DOI 10.1007/BF02756711
- Pierre Samuel, Compléments à un article de Hans Grauert sur la conjecture de Mordell, Inst. Hautes Études Sci. Publ. Math. 29 (1966), 55–62 (French). MR 204430, DOI 10.1007/BF02684805
Additional Information
- E. Hrushovski
- Affiliation: Department of Mathematics, Hebrew University, Jerusalem, Israel
- Email: ehud@sunset.ma.huji.ac.il
- M. Itai
- Affiliation: Department of Mathematical Sciences, Tokai University, Hiratsuka 259-1292, Japan
- Email: itai@ss.u-tokai.ac.jp
- Received by editor(s): August 1, 1998
- Published electronically: July 8, 2003
- Additional Notes: The first author thanks Miller Institute at the University of California, Berkeley
- © Copyright 2003 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 355 (2003), 4267-4296
- MSC (2000): Primary 03C60, 12H05
- DOI: https://doi.org/10.1090/S0002-9947-03-03264-1
- MathSciNet review: 1990753