The Journal of Symbolic Logic

Research Article

Pfaffian differential equations over exponential o-minimal structures

Chris Millera1 * and Patrick Speisseggera2 c1

a1 Department of Mathematics, University of Toronto, Toronto, Ontario, Canada, M5S 3G3

a2 Department of Mathematics, The Ohio State University, 231 West 18Th Avenue, Columbus, Ohio 43210, USA, E-mail:, URL:

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 = (F 1,…, F 1): (a, b) → ℝ i is ℜ-Pfaffian if there exists G: ℝ1+l → ℝ l 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) → ℝ l is ℜ-Pfaffian, then (ℜ, F) is o-minimal (Proposition 7). We say that F: ℝ → ℝ l 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 lim t→+∞[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.

(Received September 13 2000)

(Revised April 24 2001)


c1 Department of Mathematics, University of Wisconsin, 480 Lincoln Drive, Madison, Wisconsin 53706, USA, E-mail:, URL:


*   Research supported by NSF Grants DMS-9896225 and DMS-9988855.

  Research supported by NSERC Grant OGP0009070 and in part by NSF Grant DMS-9988453.