Submission date: 1 March 2018

Abstract:

Given a model of the theory of the real field with restricted analytic functions such that its value group has finite archimedean rank we show how one can extend the restricted logarithm to a global logarithm with values in the polynomial ring over the model with dimension the archimedean rank. The logarithms are determined by algebraic data from the model, namely by a section of the model and by an embedding of the value group into its Hahn group.
If the archimedean rank of the value group coincides with the rational rank the logarithms are equivalent.
We illustrate how one can embed such a logarithm into a model of the real field with restricted analytic functions and exponential. This allows us to define constructible functions with good lifting properties. As an application we establish a Lebesgue measure and integration theory with values in the polynomial ring, extending and strengthening the construction in [T. Kaiser: Lebesgue measure and integration theory on non-archimedean real closed fields with archimedean value group. Proc. Lond. Math. Soc. 116 (2018), no. 2, 209-247.].

Mathematics Subject Classification: 03C64, 03H05, 06F20, 12J25, 26E30, 28B15, 28E05, 32B20

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Full text arXiv 1803.00598: pdf, ps.