Lengths of developments in K((G))

• Julia F. Knight ^[1] ; Karen Lange ^[2]
  1. [1] University of Notre Dame

     University of Notre Dame

     Township of Portage, Estados Unidos

     
  2. [2] Wellesley College

     Wellesley College

     Town of Wellesley, Estados Unidos

     
• Localización: Selecta Mathematica, New Series, ISSN 1022-1824, Vol. 25, Nº. 1, 2019
• Idioma: inglés
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• Resumen

  • 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 ≺ 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 ≺ has order type ω , then all elements of R have length less than ωωω , 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 ω , then the elements have length less than ωωω . 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 γ , where ω≤γ<ω1 , then the elements of R have length at most ωωγ .