Differential equations over polynomially bounded o-minimal structures
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- by Jean-Marie Lion, Chris Miller and Patrick Speissegger PDF
- Proc. Amer. Math. Soc. 131 (2003), 175-183 Request permission
Abstract:
We investigate the asymptotic behavior at $+\infty$ of non-oscillatory solutions to differential equations $y’=G(t,y), t>a$, where $G\colon \mathbb {R}^{1+l}\to \mathbb {R}^l$ is definable in a polynomially bounded o-minimal structure. In particular, we show that the Pfaffian closure of a polynomially bounded o-minimal structure on the real field is levelled.References
- Edward Bierstone and Pierre D. Milman, Semianalytic and subanalytic sets, Inst. Hautes Études Sci. Publ. Math. 67 (1988), 5–42. MR 972342
- Michael Boshernitzan, Universal formulae and universal differential equations, Ann. of Math. (2) 124 (1986), no. 2, 273–291. MR 855296, DOI 10.2307/1971279
- F. Cano, R. Moussu, and F. Sanz, Oscillation, spiralement, tourbillonnement, Comment. Math. Helv. 75 (2000), no. 2, 284–318 (French, with English and French summaries). MR 1774707, DOI 10.1007/s000140050127
- Lou van den Dries, A generalization of the Tarski-Seidenberg theorem, and some nondefinability results, Bull. Amer. Math. Soc. (N.S.) 15 (1986), no. 2, 189–193. MR 854552, DOI 10.1090/S0273-0979-1986-15468-6
- Lou van den Dries, $T$-convexity and tame extensions. II, J. Symbolic Logic 62 (1997), no. 1, 14–34. MR 1450511, DOI 10.2307/2275729
- Lou van den Dries, Tame topology and o-minimal structures, London Mathematical Society Lecture Note Series, vol. 248, Cambridge University Press, Cambridge, 1998. MR 1633348, DOI 10.1017/CBO9780511525919
- Lou van den Dries, Angus Macintyre, and David Marker, The elementary theory of restricted analytic fields with exponentiation, Ann. of Math. (2) 140 (1994), no. 1, 183–205. MR 1289495, DOI 10.2307/2118545
- Lou van den Dries and Chris Miller, Geometric categories and o-minimal structures, Duke Math. J. 84 (1996), no. 2, 497–540. MR 1404337, DOI 10.1215/S0012-7094-96-08416-1
- Lou van den Dries and Patrick Speissegger, The real field with convergent generalized power series, Trans. Amer. Math. Soc. 350 (1998), no. 11, 4377–4421. MR 1458313, DOI 10.1090/S0002-9947-98-02105-9
- —, The field of reals with multisummable series and the exponential function, Proc. London Math. Soc. 81 (2000), 513–565.
- Franz-Viktor Kuhlmann and Salma Kuhlmann, The exponential rank of nonarchimedean exponential fields, Real algebraic geometry and ordered structures (Baton Rouge, LA, 1996) Contemp. Math., vol. 253, Amer. Math. Soc., Providence, RI, 2000, pp. 181–201. MR 1747584, DOI 10.1090/conm/253/03931
- Jean-Marie Lion, Inégalité de Lojasiewicz en géométrie pfaffienne, Illinois J. Math. 44 (2000), no. 4, 889–900 (French, with English summary). MR 1804312
- Jean-Marie Lion and Patrick Speissegger, Analytic stratification in the Pfaffian closure of an o-minimal structure, Duke Math. J. 103 (2000), no. 2, 215–231. MR 1760626, DOI 10.1215/S0012-7094-00-10322-5
- David Marker and Chris Miller, Levelled o-minimal structures, Rev. Mat. Univ. Complut. Madrid 10 (1997), no. Special Issue, suppl., 241–249. Real algebraic and analytic geometry (Segovia, 1995). MR 1485302
- Chris Miller, Expansions of the real field with power functions, Ann. Pure Appl. Logic 68 (1994), no. 1, 79–94. MR 1278550, DOI 10.1016/0168-0072(94)90048-5
- Chris Miller, Exponentiation is hard to avoid, Proc. Amer. Math. Soc. 122 (1994), no. 1, 257–259. MR 1195484, DOI 10.1090/S0002-9939-1994-1195484-5
- Chris Miller, Infinite differentiability in polynomially bounded o-minimal structures, Proc. Amer. Math. Soc. 123 (1995), no. 8, 2551–2555. MR 1257118, DOI 10.1090/S0002-9939-1995-1257118-1
- Chris Miller, A growth dichotomy for o-minimal expansions of ordered fields, Logic: from foundations to applications (Staffordshire, 1993) Oxford Sci. Publ., Oxford Univ. Press, New York, 1996, pp. 385–399. MR 1428013
- C. Miller and P. Speissegger, Pfaffian differential equations over exponential o-minimal structures, J. Symbolic Logic 67 (2002), 438–448.
- J.-P. Rolin, P. Speissegger, and A. Wilkie, Quasianalytic Denjoy-Carleman classes and o-minimality, preprint (2001).
- Maxwell Rosenlicht, The rank of a Hardy field, Trans. Amer. Math. Soc. 280 (1983), no. 2, 659–671. MR 716843, DOI 10.1090/S0002-9947-1983-0716843-5
- Maxwell Rosenlicht, Rank change on adjoining real powers to Hardy fields, Trans. Amer. Math. Soc. 284 (1984), no. 2, 829–836. MR 743747, DOI 10.1090/S0002-9947-1984-0743747-5
- Maxwell Rosenlicht, Growth properties of functions in Hardy fields, Trans. Amer. Math. Soc. 299 (1987), no. 1, 261–272. MR 869411, DOI 10.1090/S0002-9947-1987-0869411-2
- John Shackell, Rosenlicht fields, Trans. Amer. Math. Soc. 335 (1993), no. 2, 579–595. MR 1085945, DOI 10.1090/S0002-9947-1993-1085945-5
- Patrick Speissegger, The Pfaffian closure of an o-minimal structure, J. Reine Angew. Math. 508 (1999), 189–211. MR 1676876, DOI 10.1515/crll.1999.026
Additional Information
- Jean-Marie Lion
- Affiliation: Laboratoire de Topologie, Université de Bourgogne, 21078 Dijon cedex, France
- Address at time of publication: IRMAR, Campus Beaulieu, Université Rennes I, 35042 Rennes cedex, France
- Email: lion@maths.univ-rennes1.fr
- Chris Miller
- Affiliation: Department of Mathematics, The Ohio State University, 231 West 18th Avenue, Columbus, Ohio 43210
- MR Author ID: 330760
- Email: miller@math.ohio-state.edu
- Patrick Speissegger
- Affiliation: Department of Mathematics, University of Toronto, Toronto, Ontario, Canada M5S 3G3
- Address at time of publication: Department of Mathematics, University of Wisconsin, 480 Lincoln Drive, Madison, Wisconsin 53706
- MR Author ID: 361060
- Email: speisseg@math.wisc.edu
- Received by editor(s): April 11, 2000
- Received by editor(s) in revised form: August 20, 2001
- Published electronically: May 22, 2002
- Additional Notes: The second author’s research was supported by NSF Grants DMS-9896225 and DMS-9988855.
The third author’s research was supported in part by NSERC Grant OGP0009070. - Communicated by: Carl G. Jockusch, Jr.
- © Copyright 2002 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 131 (2003), 175-183
- MSC (2000): Primary 26A12, 34E99; Secondary 34E05, 03C64
- DOI: https://doi.org/10.1090/S0002-9939-02-06509-7
- MathSciNet review: 1929037