广义级数域上的Hardy型导子
Hardy type derivations on generalized series fields
摘要：
本文考虑广义级数的值域$\mathds{K}:=\mathbb{R}((Γ))$（系数为实且单项式为全序乘法群$Γ$）.研究如何赋予$\mathds{K}$一个级数导数，即满足无穷和可交换（强线性）和莱布尼茨规则（的无穷版本）等自然性质的导数.我们的特点时，这样的衍生物是哈代型，也就是说，当它表现得像分化芽的实值函数在哈代域。本文给出了Hardy型导子是满射的一个充要条件。
英文摘要：
We consider the valued field $\mathds{K}:=\mathbb{R}((Γ))$ of generalized series (with real coefficients and monomials in a totally ordered multiplicative group $Γ$). We investigate how to endow $\mathds{K}$ with a series derivation, that is a derivation that satisfies some natural properties such as commuting with infinite sums (strong linearity) and (an infinite version of) Leibniz rule. We characterize when such a derivation is of Hardy type, that is, when it behaves like differentiation of germs of real valued functions in a Hardy field. We provide a necessary and sufficent condition for a series derivation of Hardy type to be surjective.
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