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arXiv · 1010.0896

Hardy type derivations on fields of exponential logarithmic series

Salma Kuhlmann·Mickael Matusinski
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Abstract

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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Salma Kuhlmann, Mickael Matusinski. 2011-09-11. Hardy type derivations on fields of exponential logarithmic series. https://doi.org/10.1016/j.jalgebra.2011.07.023

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