The main goal of this note is to study for certain o-minimal structures the following propriety: for each definable C∞ function g0: [0, 1] → ℝ there is a definable C∞ function g: [–ε, 1] → ℝ, for some ε > 0, such that g = g0 for all x ∈ [0, 1].

We show that the ordered field of real numbers with restricted $\mathbb{R}_{\mathscr{H}}$-definable analytic functions admits quantifier elimination if we add a function symbol $^{-1}$ for the function $x\mapsto \frac{1}{x}$ (with $0^{-1}=0$ by convention), where $\mathbb{R}_{\mathscr{H}}$ is the real field augmented by the functions in the family $\mathscr{H}$ of restricted parts (real and imaginary) of holomorphic functions which satisfies certain conditions. Further, with another condition on $\mathscr{H}$ we show that the structure ($\mathbb{R}_{\mathscr{H}}$, constants) is strongly model complete.

Let ℛ be an o-minimal expansion of a real closed field R. We continue here the investigation we began in [11] of differentiability with respect to the algebraically closed field [Formula: see text]. We develop the basic theory of such K-differentiability for definable functions of several variables, proving theorems on removable singularities as well as analogues of the Weierstrass preparation and division theorems for definable functions. We consider also definably meromorphic functions and prove that every definable function which is meromorphic (...) on Kn is necessarily a rational function. We finally discuss definable analogues of complex analytic manifolds, with possible connections to the model theoretic work on compact complex manifolds, and present two examples of "nonstandard manifolds" in our setting. (shrink)

We investigate expansions of the ordered field of real numbers equipped with a family of real power functions. We show in particular that the theory of the ordered field of real numbers augmented by all restricted analytic functions and all real power functions admits elimination of quantifiers and has a universal axiomatization. We derive that every function of one variable definable in this structure, not ultimately identically 0, is asymptotic at + ∞ to a real function of the form x (...) cxr, c ≠ 0; in particular, this structure is polynomially bounded. Furthermore, given any definable function f:U → with U open in n, if α ε U and f is infinitely differentiable at α, then f is real analytic in a neighborhood of α. (shrink)

In 2001, the algebraico-tree-theoretic simplicity hierarchical structure of J. H. Conway’s ordered field ${\mathbf {No}}$ of surreal numbers was brought to the fore by the first author and employed to provide necessary and sufficient conditions for an ordered field to be isomorphic to an initial subfield of ${\mathbf {No}}$, i.e. a subfield of ${\mathbf {No}}$ that is an initial subtree of ${\mathbf {No}}$. In this sequel, analogous results are established for ordered exponential fields, making use of a slight generalization of (...) Schmeling’s conception of a transseries field. It is further shown that a wide range of ordered exponential fields are isomorphic to initial exponential subfields of $$. These include all models of $T$, where ${\mathbb R}_W$ is the reals expanded by a convergent Weierstrass system W. Of these, those we call trigonometric-exponential fields are given particular attention. It is shown that the exponential functions on the initial trigonometric-exponential subfields of ${\mathbf {No}}$, which includes ${\mathbf {No}}$ itself, extend to canonical exponential functions on their surcomplex counterparts. The image of the canonical map of the ordered exponential field ${\mathbb T}^{LE}$ of logarithmic-exponential transseries into ${\mathbf {No}}$ is shown to be initial, as are the ordered exponential fields ${\mathbb R})^{EL}$ and ${\mathbb R}\langle \langle \omega \rangle \rangle $. (shrink)

We prove the model completeness of expansions of the reals by restricted elliptic and abelian functions. We make use of an auxiliary structure admitting quantifier elimination, where the basic relations are strongly definable in the original structure.