Pfaffian differential equations over exponential o-minimal structures

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 : ℝ 1+ l → ℝ l definable in ℜ such that F ′( t ) = G ( t, F ( t )) for all t ∈ ( a, b ) and each component function G i : ℝ 1+ l → ℝ is independent of the last l − i 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 e x : ℝ → ℝ (Proposition 2). Note that the above definitions make sense for expansions of arbitrary ordered fields.