指数对数级数域上的Hardy型导子
Hardy type derivations on fields of exponential logarithmic series
摘要：
本文考虑形式级数的值域$\mathds{K}:=\mathbb{R}((Γ))$（系数为实且单项式为全序乘法群$Γ>$）.研究了如何赋予$\mathds{K}$一个对数$l$，使其满足与单项式的无穷乘积可交换等自然性质.在“广义级数域上的Hardy型导子”一文中，我们研究了$\mathds{K}$上的导子.在这里，我们调查的相容性条件之间的对数和导数，即当对数导数是对数的导数。我们分析了一个给定的导数的充分条件，通过集成的对数导数构建一个兼容的对数。在她的专著“有序指数场”中，第一作者描述了$(\mathds{K},l)$的指数闭包$\mathds{K}^{\rm{EL}}$。在这里，我们将展示如何将$\mathds{K}$上的这种日志兼容推导扩展到$\mathds{K}^{\rm{EL}}$。
英文摘要：
We consider the valued field $\mathds{K}:=\mathbb{R}((Γ))$ of formal series (with real coefficients and monomials in a totally ordered multiplicative group $Γ>$). We investigate how to endow $\mathds{K}$ with a logarithm $l$, which satisfies some natural properties such as commuting with infinite products of monomials. In the article "Hardy type derivations on generalized series fields", we study derivations on $\mathds{K}$. Here, we investigate compatibility conditions between the logarithm and the derivation, i.e. when the logarithmic derivative is the derivative of the logarithm. We analyse sufficient conditions on a given derivation to construct a compatible logarithm via integration of logarithmic derivatives. In her monograph "Ordered exponential fields", the first author described the exponential closure $\mathds{K}^{\rm{EL}}$ of $(\mathds{K},l)$. Here we show how to extend such a log-compatible derivation on $\mathds{K}$ to $\mathds{K}^{\rm{EL}}$.
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