Hostname: page-component-7c8c6479df-r7xzm Total loading time: 0 Render date: 2024-03-29T11:45:25.976Z Has data issue: false hasContentIssue false

Pfaffian differential equations over exponential o-minimal structures

Published online by Cambridge University Press:  12 March 2014

Chris Miller
Affiliation:
Department of Mathematics, University of Toronto, Toronto, Ontario, Canada, M5S 3G3
Patrick Speissegger*
Affiliation:
Department of Mathematics, The Ohio State University, 231 West 18Th Avenue, Columbus, Ohio 43210, USA, E-mail: miller@math.ohio-state.edu, URL: http://www.math.ohio-state.edu/~miller
*
Department of Mathematics, University of Wisconsin, 480 Lincoln Drive, Madison, Wisconsin 53706, USA, E-mail: speisseg@math.wisc.edu, URL: http://www.math.wisc.edu/~speisseg

Extract

In this paper, we continue investigations into the asymptotic behavior of solutions of differential equations over o-minimal structures.

Let ℜ be an expansion of the real field (ℝ, +, ·).

A differentiable map F = (F1,…, F1): (a, b) → ℝi is ℜ-Pfaffian if there exists G: ℝ1+l → ℝl definable in ℜ such that F′(t) = G(t, F(t)) for all t ∈ (a, b) and each component function Gi: ℝ1+l → ℝ is independent of the last li variables (i = 1, …, l). If ℜ is o-minimal and F: (a, b) → ℝl is ℜ-Pfaffian, then (ℜ, F) is o-minimal (Proposition 7). We say that F: ℝ → ℝl is ultimately ℜ-Pfaffian if there exists r ∈ ℝ such that the restriction F ↾(r, ∞) is ℜ-Pfaffian. (In general, ultimately abbreviates “for all sufficiently large positive arguments”.)

The structure ℜ is closed under asymptotic integration if for each ultimately non-zero unary (that is, ℝ → ℝ) function f definable in ℜ there is an ultimately differentiable unary function g definable in ℜ such that limt→+∞[g′(t)/f(t)] = 1- If ℜ is closed under asymptotic integration, then ℜ is o-minimal and defines ex: ℝ → ℝ (Proposition 2).

Note that the above definitions make sense for expansions of arbitrary ordered fields.

Type
Research Article
Copyright
Copyright © Association for Symbolic Logic 2002

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1]Aschenbrenner, M. and van den Dries, L., Closed asymptotic couples, Journal of Algebra, vol. 225 (2000), pp. 309358.CrossRefGoogle Scholar
[2]Lion, J.-M., Miller, C., and Speissegger, P., Differential equations over polynomially bounded o-minimal structures, Proceedings of the American Mathematical Society, (to appear).Google Scholar
[3]Miller, C., Exponentiation is hard to avoid, Proceedings of the American Mathematical Society, vol. 122 (1994), pp. 257259.CrossRefGoogle Scholar
[4]Miller, C., A growth dichotomy for o-minimal expansions of ordered fields, Logic: From foundations to applications, Oxford Science Publications, Oxford University Press, 1996, pp. 385399.CrossRefGoogle Scholar
[5]Miller, C. and Starchenko, S., A growth dichotomy for o-minimal expansions of ordered groups, Transactions of the American Mathematical Society, vol. 350 (1998), pp. 35053521.CrossRefGoogle Scholar
[6]Otero, M., Peterzil, Y., and Pillay, A., On groups and rings definable in o-minimal expansions of real closed fields, Bulletin of the London Mathematical Society, vol. 28 (1996), pp. 714.CrossRefGoogle Scholar
[7]Peterzil, Y., Speissegger, P., and Starchenko, S., Adding multiplication to an o-minimal expansion of the additive group of real numbers, Logic colloquium 98, Lecture Notes in Logic, vol. 13, A. K. Peters, 2000, pp. 357362.Google Scholar
[8]Rosenlicht, M., The rank of a Hardy field, Transactions of the American Mathematical Society, vol. 280 (1983), pp. 659671.CrossRefGoogle Scholar
[9]Rosenlicht, M., Growth properties of functions in Hardy fields, Transactions of the American Mathematical Society, vol. 299 (1987), pp. 261272.CrossRefGoogle Scholar
[10]Speissegger, P., The Pfaffian closure of an o-minimal structure, Journal fÜr die Reine und Angewandte Mathematik, vol. 508 (1999), pp. 189211.CrossRefGoogle Scholar
[11]van den Dries, L., o-Minimal structures, Logic: From foundations to applications, Oxford Science Publications, Oxford University Press, New York, 1996, pp. 137185.CrossRefGoogle Scholar
[12]van den Dries, L., Tame topology and o-minimal structures, London Mathematical Society Lecture Note Series, vol. 248, Cambridge University Press, Cambridge, 1998.CrossRefGoogle Scholar
[13]van den Dries, L., Macintyre, A., and Marker, D., The elementary theory of restricted analytic fields with exponentiation, Annals of Mathematics, vol. 140 (1994), pp. 183205.CrossRefGoogle Scholar
[14]van den Dries, L., Macintyre, A., and Marker, D., Logarithmic-exponential power series, Journal of the London Mathematical Society, vol. 56 (1997), pp. 417434.CrossRefGoogle Scholar
[15]van den Dries, L. and Miller, C., Geometric categories and o-minimal structures, Duke Mathematical Journal, vol. 84 (1996), pp. 497540.CrossRefGoogle Scholar