Residue field domination in henselian valued fields of equicharacteristic zero

Abstract

In this paper we investigate domination results for henselian valued fields of equicharacteristic zero for elements in the home sort. We introduce a notion of weak opaqueness to construct resolutions for certain examples of valued fields, that include among others the theory of the Laurent series over the complex numbers.We follow ideas present in [2], and some results are present in their proofs but not in an easy quotable form. Hence we include the statements and their proofs for sake of completeness.Aknowledgements: This document was written as part of the author's PhD thesis under Pierre Simon and Thomas Scanlon. The author would like to express her gratitude to both of them for many insightful conversations. The author would like as well to thank Clifton Ealy, for answering some questions in a kind and organized way when the author was learning many of the contents related to this work.

Beautiful pairs

“…After setting up the notation and terminology on valued fields that will be used for the rest of the paper, we will prove some key domination results building on and generalizing [20,Proposition 12.11] and [15, Theorem 2.5] (see subsection 6.3). Some of these results have been recently obtained independently by M. Vicaria in [44]. Slight variants also appear in [22] by the second author and R. Mennuni.…”