Hardy type derivations on fields of exponential logarithmic series

Details Hardy type derivations on fields of exponential logarithmic series Hardy type derivations on fields of exponential logarithmic series

2010-10-05

arxiv.org/abs/1010.0896v2

Journal of Algebra, Volume 345, Issue 1, 1 November 2011, Pages 171-189

Mathematics

Commutative Algebra

25 pages

Scientific paper

10.1016/j.jalgebra.2011.07.023

We consider the valued field $\mathds{K}:=\mathbb{R}((\Gamma))$ of formal series (with real coefficients and monomials in a totally ordered multiplicative group $\Gamma>$). 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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