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

TL;DR

This paper extends the concept of logarithms and constructs a measure and integration theory on non-archimedean models of the real field with restricted analytic functions, based on algebraic data and value group properties.

Contribution

It introduces a method to extend restricted logarithms to global ones in models with finite archimedean rank, enabling a new integration theory with polynomial ring values.

Findings

Extended restricted logarithms to global logarithms in models with finite archimedean rank.

Constructed a Lebesgue measure and integration theory with polynomial ring values.

Demonstrated embedding of logarithms into models with exponentiation for better lifting properties.

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 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: 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.].