Pfaffian Differential Equations over Exponential O-Minimal Structures

R l definable in R such that F (t) = G(t, F (t)) for all t ∈ (a, b) and each component function Gi : 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 is R-Pfaffian, then (R, F ) is o-minimal (Proposition 7). We say that F : R → R 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 nonzero 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 e : R → R (Proposition 2). Note that the above definitions make sense for expansions of arbitrary ordered fields.