A geometric model of an arbitrary differentially closed field of characteristic zero

Spodzieja, Stanisław

doi:10.1090/tran/8257

A geometric model of an arbitrary differentially closed field of characteristic zero
HTML articles powered by AMS MathViewer

by Stanisław Spodzieja PDF

Trans. Amer. Math. Soc. 374 (2021), 663-686 Request permission

Abstract:

We give an elementary construction of an arbitrary differentially closed field and of a universal extension of a differential field in terms of Nash function fields. We also give a characterization of any Archimedean ordered differentially closed field in terms of Nash functions.

References

Saugata Basu, Richard Pollack, and Marie-Françoise Roy, Algorithms in real algebraic geometry, Algorithms and Computation in Mathematics, vol. 10, Springer-Verlag, Berlin, 2003. MR 1998147, DOI 10.1007/978-3-662-05355-3
Riccardo Benedetti and Jean-Jacques Risler, Real algebraic and semi-algebraic sets, Actualités Mathématiques. [Current Mathematical Topics], Hermann, Paris, 1990. MR 1070358
Lenore Blum, Differentially closed fields: a model-theoretic tour, Contributions to algebra (collection of papers dedicated to Ellis Kolchin), Academic Press, New York, 1977, pp. 37–61. MR 0491149
J. Bochnak, M. Coste, and M.-F. Roy, Géométrie algébrique réelle, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 12, Springer-Verlag, Berlin, 1987 (French). MR 949442
Jacek Bochnak and Gustave Efroymson, Real algebraic geometry and the 17th Hilbert problem, Math. Ann. 251 (1980), no. 3, 213–241. MR 589251, DOI 10.1007/BF01428942
Ludwig Bröcker, Real spectra and distributions of signatures, Real algebraic geometry and quadratic forms (Rennes, 1981) Lecture Notes in Math., vol. 959, Springer, Berlin-New York, 1982, pp. 249–272. MR 683137
Patrizia Gianni, Barry Trager, and Gail Zacharias, Gröbner bases and primary decomposition of polynomial ideals, J. Symbolic Comput. 6 (1988), no. 2-3, 149–167. Computational aspects of commutative algebra. MR 988410, DOI 10.1016/S0747-7171(88)80040-3
Gert-Martin Greuel and Gerhard Pfister, A Singular introduction to commutative algebra, Second, extended edition, Springer, Berlin, 2008. With contributions by Olaf Bachmann, Christoph Lossen and Hans Schönemann; With 1 CD-ROM (Windows, Macintosh and UNIX). MR 2363237
Leo Harrington, Recursively presentable prime models, J. Symbolic Logic 39 (1974), 305–309. MR 351804, DOI 10.2307/2272643
J. van der Hoeven, Transseries and real differential algebra, Lecture Notes in Mathematics, vol. 1888, Springer-Verlag, Berlin, 2006. MR 2262194, DOI 10.1007/3-540-35590-1
Irving Kaplansky, An introduction to differential algebra, Publ. Inst. Math. Univ. Nancago, No. 5, Hermann, Paris, 1957. MR 0093654
E. R. Kolchin, Differential algebra and algebraic groups, Pure and Applied Mathematics, Vol. 54, Academic Press, New York-London, 1973. MR 0568864
E. R. Kolchin, On universal extensions of differential fields, Pacific J. Math. 86 (1980), no. 1, 139–143. MR 586873, DOI 10.2140/pjm.1980.86.139
Kazimierz Kuratowski and Andrzej Mostowski, Set theory, Second, completely revised edition, Studies in Logic and the Foundations of Mathematics, Vol. 86, North-Holland Publishing Co., Amsterdam-New York-Oxford; PWN—Polish Scientific Publishers, Warsaw, 1976. With an introduction to descriptive set theory; Translated from the 1966 Polish original. MR 0485384
Stanisław Łojasiewicz, Introduction to complex analytic geometry, Birkhäuser Verlag, Basel, 1991. Translated from the Polish by Maciej Klimek. MR 1131081, DOI 10.1007/978-3-0348-7617-9
David Marker, Model theory, Graduate Texts in Mathematics, vol. 217, Springer-Verlag, New York, 2002. An introduction. MR 1924282
David Mumford, Algebraic geometry. I, Grundlehren der Mathematischen Wissenschaften, No. 221, Springer-Verlag, Berlin-New York, 1976. Complex projective varieties. MR 0453732
John Nash, Real algebraic manifolds, Ann. of Math. (2) 56 (1952), 405–421. MR 50928, DOI 10.2307/1969649
David Pierce and Anand Pillay, A note on the axioms for differentially closed fields of characteristic zero, J. Algebra 204 (1998), no. 1, 108–115. MR 1623945, DOI 10.1006/jabr.1997.7359
Alexander Prestel and Charles N. Delzell, Positive polynomials, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2001. From Hilbert’s 17th problem to real algebra. MR 1829790, DOI 10.1007/978-3-662-04648-7
Joseph Fels Ritt, Differential equations from the algebraic standpoint, American Mathematical Society Colloquium Publications, vol. 14, American Mathematical Society, New York, 1932. MR 3822122, DOI 10.1090/coll/014
Joseph Fels Ritt, Differential algebra, Dover Publications, Inc., New York, 1966. MR 0201431
J. F. Ritt and E. R. Kolchin, On certain ideals of differential polynomials, Bull. Amer. Math. Soc. 45 (1939), 895–898. MR 606, DOI 10.1090/S0002-9904-1939-07112-7
Lorenzo Robbiano and Moss Sweedler, Ideal and subalgebra coefficients, Proc. Amer. Math. Soc. 126 (1998), no. 8, 2213–2219. MR 1443407, DOI 10.1090/S0002-9939-98-04306-8
Abraham Robinson, Complete theories, North-Holland Publishing Co., Amsterdam, 1956. MR 0075897
Abraham Robinson, On the concept of a differentially closed field, Bull. Res. Council Israel Sect. F 8F (1959), 113–128 (1959). MR 125016
Abraham Robinson, Local differential algebra, Trans. Amer. Math. Soc. 97 (1960), 427–456. MR 122815, DOI 10.1090/S0002-9947-1960-0122815-9
Abraham Robinson, Ordered differential fields, J. Combinatorial Theory Ser. A 14 (1973), 324–333. MR 323768, DOI 10.1016/0097-3165(73)90009-5
Gerald E. Sacks, The differential closure of a differential field, Bull. Amer. Math. Soc. 78 (1972), 629–634. MR 299466, DOI 10.1090/S0002-9904-1972-12969-0
A. Seidenberg, Some basic theorems in differential algebra (characteristic $p$, arbitrary), Trans. Amer. Math. Soc. 73 (1952), 174–190. MR 49174, DOI 10.1090/S0002-9947-1952-0049174-X
A. Seidenberg, A new decision method for elementary algebra, Ann. of Math. (2) 60 (1954), 365–374. MR 63994, DOI 10.2307/1969640
A. Seidenberg, Abstract differential algebra and the analytic case, Proc. Amer. Math. Soc. 9 (1958), 159–164. MR 93655, DOI 10.1090/S0002-9939-1958-0093655-0
A. Seidenberg, Abstract differential algebra and the analytic case. II, Proc. Amer. Math. Soc. 23 (1969), 689–691. MR 248122, DOI 10.1090/S0002-9939-1969-0248122-5
Saharon Shelah, Differentially closed fields, Israel J. Math. 16 (1973), 314–328. MR 344116, DOI 10.1007/BF02756711
Michael F. Singer, The model theory of ordered differential fields, J. Symbolic Logic 43 (1978), no. 1, 82–91. MR 495120, DOI 10.2307/2271951
Michael F. Singer, A class of differential fields with minimal differential closures, Proc. Amer. Math. Soc. 69 (1978), no. 2, 319–322. MR 465851, DOI 10.1090/S0002-9939-1978-0465851-4
Stanisław Spodzieja, A geometric model of an arbitrary real closed field, Pacific J. Math. 264 (2013), no. 2, 455–469. MR 3089403, DOI 10.2140/pjm.2013.264.455
Stanisław Spodzieja, The field of Nash functions and factorization of polynomials, Ann. Polon. Math. 65 (1996), no. 1, 81–94. MR 1414752, DOI 10.4064/ap-65-1-81-94
Alfred Tarski, A Decision Method for Elementary Algebra and Geometry, The Rand Corporation, Santa Monica, Calif., 1948. MR 0028796