kappa-bounded Exponential-Logarithmic Power Series Fields

Comparison of Exponential-Logarithmic and Logarithmic-Exponential series

We explain how the field of logarithmic-exponential series constructed in \cite{DMM1} and \cite {DMM2} embeds as an exponential field in any field of exponential-logarithmic series constructed in \cite{KK1}, \cite {K} and \cite {KS}. On the other hand, we explain why no field of exponential-logarithmic series embeds in the field of logarithmic-exponential series. This clarifies why the two constructions are intrinsically different, in the sense that they produce non-isomorphic models of Th$(\R_{an, exp})$; the elementary theory of the ordered field of real numbers, with the exponential functio…

A Note on Schanuel's Conjectures for Exponential Logarithmic Power Series Fields

In [1], J. Ax proved a transcendency theorem for certain differential fields of characteristic zero: the differential counterpart of the still open Schanuel's conjecture about the exponential function over the field of complex numbers [11, page 30]. In this article, we derive from Ax's theorem transcendency results in the context of differential valued exponential fields. In particular, we obtain results for exponential Hardy fields, Logarithmic-Exponential power series fields and Exponential-Logarithmic power series fields.

The exponential-logarithmic equivalence classes of surreal numbers

In his monograph, H. Gonshor showed that Conway's real closed field of surreal numbers carries an exponential and logarithmic map. Subsequently, L. van den Dries and P. Ehrlich showed that it is a model of the elementary theory of the field of real numbers with the exponential function. In this paper, we give a complete description of the exponential equivalence classes in the spirit of the classical Archimedean and multiplicative equivalence classes. This description is made in terms of a recursive formula as well as a sign sequence formula for the family of representatives of minimal length …

Surreal fields stable under exponential and logarithmic functions

Surreal numbers, have a very rich and elegant theory. This class of numbers, denoted by No, includes simultaneously the ordinal numbers and the real numbers, and forms a universal huge real closed field: It is universal in the sense that any real closed field can be embedded in it. Following Gonshor, surreal numbers can also be seen as signs sequences of ordinal length, with some exponential and logarithmic functions that extend the usual functions over the reals. No can actually also be seen as an elegant particular (generalized) power series field with real coefficients, namely Hahn series w…

Surreal fields stable under exponential, logarithmic, derivative and anti-derivative functions

The class of surreal numbers, denoted by $\textbf{No}$, initially proposed by Conway, is a universal ordered field in the sense that any ordered field can be embedded in it. They include in particular the real numbers and the ordinal numbers. They have strong relations with other fields such as field of transseries. Following Gonshor, surreal numbers can be seen as signs sequences of ordinal length, with some exponential and logarithmic functions that extend the usual functions over the reals. $\textbf{No}$ can actually be seen as an elegant (generalized) power series field with real coefficie…

尖锐 o-极小结构中的对数-指数准备

Logarithmic--exponential preparation in sharply o-minimal structures

本文通过复解析方法，为实数无限制指数结构建立了尖锐 o-极小性、Wilkie 定理及猜想的尖锐形式，并推广了有理点插值定理。