Differential equations over polynomially bounded o-minimal structures

Lion, Jean-Marie; Miller, Chris; Speissegger, Patrick

doi:10.1090/S0002-9939-02-06509-7

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Differential equations over polynomially bounded o-minimal structures

Authors: Jean-Marie Lion, Chris Miller and Patrick Speissegger
Journal: Proc. Amer. Math. Soc. 131 (2003), 175-183
MSC (2000): Primary 26A12, 34E99; Secondary 34E05, 03C64
Posted: May 22, 2002
MathSciNet review: 1929037
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Abstract: We investigate the asymptotic behavior at $+\infty$ of non-oscillatory solutions to differential equations , 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.

1. Edward Bierstone and Pierre D. Milman, Semianalytic and subanalytic sets, Inst. Hautes Études Sci. Publ. Math. 67 (1988), 5–42. MR 972342 (89k:32011)
2. Michael Boshernitzan, Universal formulae and universal differential equations, Ann. of Math. (2) 124 (1986), no. 2, 273–291. MR 855296 (88a:12007), http://dx.doi.org/10.2307/1971279
3. 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 (2001k:37034), http://dx.doi.org/10.1007/s000140050127
4. 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 (88b:03048), http://dx.doi.org/10.1090/S0273-0979-1986-15468-6
5. Lou van den Dries, 𝑇-convexity and tame extensions. II, J. Symbolic Logic 62 (1997), no. 1, 14–34. MR 1450511 (98h:03048), http://dx.doi.org/10.2307/2275729
6. 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 (99j:03001)
7. 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 (95k:12015), http://dx.doi.org/10.2307/2118545
8. Lou van den Dries and Chris Miller, Geometric categories and o-minimal structures, Duke Math. J. 84 (1996), no. 2, 497–540. MR 1404337 (97i:32008), http://dx.doi.org/10.1215/S0012-7094-96-08416-1
9. 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 (99a:03036), http://dx.doi.org/10.1090/S0002-9947-98-02105-9
10. -, The field of reals with multisummable series and the exponential function, Proc. London Math. Soc. 81 (2000), 513-565.
11. 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 (2001d:12012), http://dx.doi.org/10.1090/conm/253/03931
12. Jean-Marie Lion, Inégalité de Lojasiewicz en géométrie pfaffienne, Illinois J. Math. 44 (2000), no. 4, 889–900 (French, with English summary). MR 1804312 (2001k:32013)
13. 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 (2001j:03076), http://dx.doi.org/10.1215/S0012-7094-00-10322-5
14. 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 (99c:03060)
15. Chris Miller, Expansions of the real field with power functions, Ann. Pure Appl. Logic 68 (1994), no. 1, 79–94. MR 1278550 (95i:03081), http://dx.doi.org/10.1016/0168-0072(94)90048-5
16. Chris Miller, Exponentiation is hard to avoid, Proc. Amer. Math. Soc. 122 (1994), no. 1, 257–259. MR 1195484 (94k:03042), http://dx.doi.org/10.1090/S0002-9939-1994-1195484-5
17. Chris Miller, Infinite differentiability in polynomially bounded o-minimal structures, Proc. Amer. Math. Soc. 123 (1995), no. 8, 2551–2555. MR 1257118 (95j:03069), http://dx.doi.org/10.1090/S0002-9939-1995-1257118-1
18. 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 (98a:03052)
19. C. Miller and P. Speissegger, Pfaffian differential equations over exponential o- minimal structures, J. Symbolic Logic 67 (2002), 438-448.
20. J.-P. Rolin, P. Speissegger, and A. Wilkie, Quasianalytic Denjoy-Carleman classes and o-minimality, preprint (2001).
21. Maxwell Rosenlicht, The rank of a Hardy field, Trans. Amer. Math. Soc. 280 (1983), no. 2, 659–671. MR 716843 (85d:12002), http://dx.doi.org/10.1090/S0002-9947-1983-0716843-5
22. Maxwell Rosenlicht, Rank change on adjoining real powers to Hardy fields, Trans. Amer. Math. Soc. 284 (1984), no. 2, 829–836. MR 743747 (85i:12008), http://dx.doi.org/10.1090/S0002-9947-1984-0743747-5
23. Maxwell Rosenlicht, Growth properties of functions in Hardy fields, Trans. Amer. Math. Soc. 299 (1987), no. 1, 261–272. MR 869411 (88b:12010), http://dx.doi.org/10.1090/S0002-9947-1987-0869411-2
24. John Shackell, Rosenlicht fields, Trans. Amer. Math. Soc. 335 (1993), no. 2, 579–595. MR 1085945 (93d:12012), http://dx.doi.org/10.1090/S0002-9947-1993-1085945-5
25. Patrick Speissegger, The Pfaffian closure of an o-minimal structure, J. Reine Angew. Math. 508 (1999), 189–211. MR 1676876 (2000j:14093), http://dx.doi.org/10.1515/crll.1999.026

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