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

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, Proc.

Periods, Power Series, and Integrated Algebraic Numbers

“…Note that in [24] Lion-Rolin preparation has been established for systems on the reals where only Weierstraß preparation is used. If our system is a convergent Analysis system we can extend the results of Lion and Rolin [23] and Comte, Lion and Rolin [3] on integration (see also Cluckers and Dan Miller [2] for an extension and [18,19] for an application to integration on non-standard real closed fields) to show that parameterized integrals in the structure of restricted power series with respect to the system are given by products of sums of definable functions and logarithm of positive definable functions. This is well-defined since the coefficient field of a convergent Analysis system is closed under logarithm.…”