Lengths of developments in K((G))

Abstract

Mourgues and Ressayre (J Symb Logic 58:641–647, 1993) showed that every real closed field F has an integer part, where this is an ordered subring with the properties appropriate for the range of a floor function. The Mourgues and Ressayre construction is canonical once we fix a residue field section K and a well ordering $\prec$ of F. The construction produces a section of the value group G of F, and a development function d mapping F isomorphically onto a truncation closed subfield R of the Hahn field K((G)). In Knight and Lange (Proc Lond Math Soc 107:177–197, 2013), the authors conjectured that if $\prec$ has order type $\omega$, then all elements of R have length less than $\omega ^{\omega ^\omega }$, and they gave examples showing that the conjectured bound would be sharp. The current paper has two theorems bounding the lengths of elements of a truncation closed subfield R of a Hahn field K((G)) in terms of the length of a “tc-basis”. Here K is a field that is either real closed or algebraically closed of characteristic 0, and G is a divisible ordered Abelian group. One theorem says that if R has a tc-basis of length at most $\omega$, then the elements have length less than $\omega ^{\omega ^\omega }$. This theorem yields the conjecture from Knight and Lange (2013). The other theorem says that if the group G is Archimedean, and R has a tc-basis of length $\gamma$, where $\omega \le \gamma < \omega _1$, then the elements of R have length at most $\omega ^{\omega ^\gamma }$.