Residue fields of arbitrary convex valuations on restricted analytic fields with exponentiation

In their paper [D–M–M2], van den Dries, Macintyre and Marker give an explicit construction of a nonarchimedean model of the theory of the reals with restricted analytic functions and exponentiation. This model, called the logarithmic exponential power series field, lies in a generalized power series field. They use the results of Ressayre and Mourgues about truncation-closed embeddings to answer a problem raised by Hardy. They also show that certain functions, including the Gamma-function and the Riemann Zeta-function, cannot be defined using exponential function, logarithm and restricted analytic functions. This paper answers a question raised by Angus Macintyre in a talk for the Algebraic Model Theory Programme at the Fields Institute, November 1996. He asked whether the results of [D–M–M2] can be deduced by a “more invariant” version of truncation. Indeed, we derive the results of [D–M–M2] without using embeddings in the logarithmic exponential power series field. We replace truncation results by an intrinsic property, which is an assertion about the residue fields of arbitrary convex valuations. It is invariant because it does not depend on an embedding in logarithmic exponential power series fields. Our result about the residue fields is proved using some comparingly simple lemmas which build on the “Valuation Property of restricted analytic functions” (cf. Corollary 3.7 of [D– M–M1]). In addition to that, we just use the knowledge of how to build up exponential fields from subfields. The following fact is well known: Take a real closed field L and a convex valuation w on L (that is, its valuation ring Ow is convex in L). Then the residue field Lw is a real closed field, and it can be embedded in L in such a way that the composition of the residue map with the embedding yields the identity on Lw. Indeed, the embedding can be constructed by use of Hensel’s Lemma (a convex valuation on a real closed field is always henselian), since the residue field has characteristic 0. The image under every such embedding is a maximal subfield of Ow , and conversely, every maximal subfield K of Ow is isomorphic to Lw via the residue map. Therefore, we will always write Lw = K ⊂ Ow , where the equality is to be understood modulo the isomorphism induced by the residue