Truncation in Differential Hahn Fields

Abstract

Being closed under truncation for subsets of generalized series fields is a robust property in the sense that it is preserved under various algebraic and transcendental extension procedures. Nevertheless, in Chapter 4 of this dissertation, we show that generalized series fields with truncation as an extra primitive yields undecidability in several settings. Our main results, however, concern the robustness of being truncation closed in generalized series fields equipped with a derivation, and under extension procedures that involve this derivation. In the last chapter, we study this in the ambient field $\mathbb{T}$ of logarithmic-exponential transseries. It leads there to a theorem saying that under a natural `splitting' condition the Liouville closure of a truncation closed differential subfield of $\mathbb{T}$ is again truncation closed.