对数跨级数域的渐近偶的NIP
NIP for the Asymptotic Couple of the Field of Logarithmic Transseries
摘要：
在对数跨级数的微分值域$\mathbb{T}_{\log}$上的求导导出了其值域群$Γ_{\log}$上的某个映射$ψ$。结构$Γ= (Γ_{\log},ψ)$是一个可分的渐近偶。在~ \cite{gehret}中，我们开始了对$(Γ_{\log},ψ)$的一阶理论的研究，其中，我们证明了理论$T_{\log} = \operatorname{Th}(Γ_{\log},ψ)$具有普遍公理化，是模型完备的，并且允许在自然的一阶语言中消除量词（QE）。在那篇论文中，我们提出了PDMATHAMT是否具有NIP的问题（即，非独立性资产）。在本文中，我们肯定地回答了这个问题：PDMATHAMT确实有NIP。我们的证明方法依赖于一个完整的调查$1$ -类型的$T_{\log}$，其中，在QE的存在下，相当于一个表征的所有简单的扩展$Γ\langleα\rangle$的$Γ$。我们还表明，$T_{\log}$不具有Steinitz交换属性，我们权衡模型之间的关系$T_{\log}$和所谓的\emph{precontraction groups}的~ \cite{kuhlmann1}。
英文摘要：
The derivation on the differential-valued field $\mathbb{T}_{\log}$ of logarithmic transseries induces on its value group $Γ_{\log}$ a certain map $ψ$. The structure $Γ= (Γ_{\log},ψ)$ is a divisible asymptotic couple. In~\cite{gehret} we began a study of the first-order theory of $(Γ_{\log},ψ)$ where, among other things, we proved that the theory $T_{\log} = \operatorname{Th}(Γ_{\log},ψ)$ has a universal axiomatization, is model complete and admits elimination of quantifiers (QE) in a natural first-order language. In that paper we posed the question whether $T_{\log}$ has NIP (i.e., the Non-Independence Property). In this paper, we answer that question in the affirmative: $T_{\log}$ does have NIP. Our method of proof relies on a complete survey of the $1$-types of $T_{\log}$, which, in the presence of QE, is equivalent to a characterization of all simple extensions $Γ\langleα\rangle$ of $Γ$. We also show that $T_{\log}$ does not have the Steinitz exchange property and we weigh in on the relationship between models of $T_{\log}$ and the so-called \emph{precontraction groups} of~\cite{kuhlmann1}.
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