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Logarithms, constructible functions and integration on non-archimedean models of the theory of the real field with restricted analytic functions with value group of finite archimedean rank

Volume 256 / 2022

Tobias Kaiser Fundamenta Mathematicae 256 (2022), 285-306 MSC: Primary 03C64; Secondary 03H05, 06F20, 12J25, 26E30, 28B15, 28E05, 32B20. DOI: 10.4064/fm70-5-2021 Published online: 2 December 2021


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 exponentiation. 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, Proc. London Math. Soc. 116 (2018), 209–247].


  • Tobias KaiserFaculty of Computer Science and Mathematics
    University of Passau
    94030 Passau, Germany

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